Calculus Examples

$x^{3} - 3 x^{2} y + 2 x y^{2} = 12$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{3} - 3 x^{2} y + 2 x y^{2}) = \frac{d}{d x} (12)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x^{3} - 3 x^{2} y + 2 x y^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 3 x^{2} y] + \frac{d}{d x} [2 x y^{2}]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 3 x^{2} y] + \frac{d}{d x} [2 x y^{2}]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [- 3 x^{2} y] + \frac{d}{d x} [2 x y^{2}]$

Step 2.2

Evaluate $\frac{d}{d x} [- 3 x^{2} y]$ .

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Step 2.2.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{2} y$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{2} y]$ .

$3 x^{2} - 3 \frac{d}{d x} [x^{2} y] + \frac{d}{d x} [2 x y^{2}]$

Step 2.2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = y$ .

$3 x^{2} - 3 (x^{2} \frac{d}{d x} [y] + y \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [2 x y^{2}]$

Step 2.2.3

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$3 x^{2} - 3 (x^{2} y' + y \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [2 x y^{2}]$

Step 2.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$3 x^{2} - 3 (x^{2} y' + y (2 x)) + \frac{d}{d x} [2 x y^{2}]$

Step 2.2.5

Move $2$ to the left of $y$ .

$3 x^{2} - 3 (x^{2} y' + 2 y x) + \frac{d}{d x} [2 x y^{2}]$

Step 2.3

Evaluate $\frac{d}{d x} [2 x y^{2}]$ .

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Step 2.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x y^{2}$ with respect to $x$ is $2 \frac{d}{d x} [x y^{2}]$ .

$3 x^{2} - 3 (x^{2} y' + 2 y x) + 2 \frac{d}{d x} [x y^{2}]$

Step 2.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = y^{2}$ .

$3 x^{2} - 3 (x^{2} y' + 2 y x) + 2 (x \frac{d}{d x} [y^{2}] + y^{2} \frac{d}{d x} [x])$

Step 2.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u$ as $y$ .

$3 x^{2} - 3 (x^{2} y' + 2 y x) + 2 (x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

Step 2.3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$3 x^{2} - 3 (x^{2} y' + 2 y x) + 2 (x (2 u \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

Step 2.3.3.3

Replace all occurrences of $u$ with $y$ .

$3 x^{2} - 3 (x^{2} y' + 2 y x) + 2 (x (2 y \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

Step 2.3.4

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$3 x^{2} - 3 (x^{2} y' + 2 y x) + 2 (x (2 y y') + y^{2} \frac{d}{d x} [x])$

Step 2.3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 x^{2} - 3 (x^{2} y' + 2 y x) + 2 (x (2 y y') + y^{2} \cdot 1)$

Step 2.3.6

Move $2$ to the left of $x$ .

$3 x^{2} - 3 (x^{2} y' + 2 y x) + 2 (2 \cdot x y y' + y^{2} \cdot 1)$

Step 2.3.7

Multiply $y^{2}$ by $1$ .

$3 x^{2} - 3 (x^{2} y' + 2 y x) + 2 (2 x y y' + y^{2})$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$3 x^{2} - 3 (x^{2} y') - 3 (2 y x) + 2 (2 x y y' + y^{2})$

Step 2.4.2

Apply the distributive property.

$3 x^{2} - 3 (x^{2} y') - 3 (2 y x) + 2 (2 x y y') + 2 y^{2}$

Step 2.4.3

Combine terms.

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Step 2.4.3.1

Multiply $2$ by $- 3$ .

$3 x^{2} - 3 x^{2} y' - 6 (y x) + 2 (2 x y y') + 2 y^{2}$

Step 2.4.3.2

Multiply $2$ by $2$ .

$3 x^{2} - 3 x^{2} y' - 6 y x + 4 x y y' + 2 y^{2}$

Step 2.4.4

Reorder terms.

$3 x^{2} + 2 y^{2} - 3 x^{2} y' - 6 x y + 4 x y y'$

Step 3

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$0$

Step 4

Reform the equation by setting the left side equal to the right side.

$3 x^{2} + 2 y^{2} - 3 x^{2} y' - 6 x y + 4 x y y' = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Move all terms not containing $y'$ to the right side of the equation.

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Step 5.1.1

Subtract $3 x^{2}$ from both sides of the equation.

$2 y^{2} - 3 x^{2} y' - 6 x y + 4 x y y' = - 3 x^{2}$

Step 5.1.2

Subtract $2 y^{2}$ from both sides of the equation.

$- 3 x^{2} y' - 6 x y + 4 x y y' = - 3 x^{2} - 2 y^{2}$

Step 5.1.3

Add $6 x y$ to both sides of the equation.

$- 3 x^{2} y' + 4 x y y' = - 3 x^{2} - 2 y^{2} + 6 x y$

Step 5.2

Factor $x y'$ out of $- 3 x^{2} y' + 4 x y y'$ .

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Step 5.2.1

Factor $x y'$ out of $- 3 x^{2} y'$ .

$x y' (- 3 x) + 4 x y y' = - 3 x^{2} - 2 y^{2} + 6 x y$

Step 5.2.2

Factor $x y'$ out of $4 x y y'$ .

$x y' (- 3 x) + x y' (4 y) = - 3 x^{2} - 2 y^{2} + 6 x y$

Step 5.2.3

Factor $x y'$ out of $x y' (- 3 x) + x y' (4 y)$ .

$x y' (- 3 x + 4 y) = - 3 x^{2} - 2 y^{2} + 6 x y$

Step 5.3

Divide each term in $x y' (- 3 x + 4 y) = - 3 x^{2} - 2 y^{2} + 6 x y$ by $x (- 3 x + 4 y)$ and simplify.

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Step 5.3.1

Divide each term in $x y' (- 3 x + 4 y) = - 3 x^{2} - 2 y^{2} + 6 x y$ by $x (- 3 x + 4 y)$ .

$\frac{x y' (- 3 x + 4 y)}{x (- 3 x + 4 y)} = \frac{- 3 x^{2}}{x (- 3 x + 4 y)} + \frac{- 2 y^{2}}{x (- 3 x + 4 y)} + \frac{6 x y}{x (- 3 x + 4 y)}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $x$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{x y' (- 3 x + 4 y)}{x (- 3 x + 4 y)} = \frac{- 3 x^{2}}{x (- 3 x + 4 y)} + \frac{- 2 y^{2}}{x (- 3 x + 4 y)} + \frac{6 x y}{x (- 3 x + 4 y)}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{y' (- 3 x + 4 y)}{- 3 x + 4 y} = \frac{- 3 x^{2}}{x (- 3 x + 4 y)} + \frac{- 2 y^{2}}{x (- 3 x + 4 y)} + \frac{6 x y}{x (- 3 x + 4 y)}$

Step 5.3.2.2

Cancel the common factor of $- 3 x + 4 y$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{y' (- 3 x + 4 y)}{- 3 x + 4 y} = \frac{- 3 x^{2}}{x (- 3 x + 4 y)} + \frac{- 2 y^{2}}{x (- 3 x + 4 y)} + \frac{6 x y}{x (- 3 x + 4 y)}$

Step 5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 3 x^{2}}{x (- 3 x + 4 y)} + \frac{- 2 y^{2}}{x (- 3 x + 4 y)} + \frac{6 x y}{x (- 3 x + 4 y)}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Cancel the common factor of $x^{2}$ and $x$ .

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Step 5.3.3.1.1.1

Factor $x$ out of $- 3 x^{2}$ .

$y' = \frac{x (- 3 x)}{x (- 3 x + 4 y)} + \frac{- 2 y^{2}}{x (- 3 x + 4 y)} + \frac{6 x y}{x (- 3 x + 4 y)}$

Step 5.3.3.1.1.2

Cancel the common factors.

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Step 5.3.3.1.1.2.1

Cancel the common factor.

$y' = \frac{x (- 3 x)}{x (- 3 x + 4 y)} + \frac{- 2 y^{2}}{x (- 3 x + 4 y)} + \frac{6 x y}{x (- 3 x + 4 y)}$

Step 5.3.3.1.1.2.2

Rewrite the expression.

$y' = \frac{- 3 x}{- 3 x + 4 y} + \frac{- 2 y^{2}}{x (- 3 x + 4 y)} + \frac{6 x y}{x (- 3 x + 4 y)}$

Step 5.3.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{3 x}{- 3 x + 4 y} + \frac{- 2 y^{2}}{x (- 3 x + 4 y)} + \frac{6 x y}{x (- 3 x + 4 y)}$

Step 5.3.3.1.3

Move the negative in front of the fraction.

$y' = - \frac{3 x}{- 3 x + 4 y} - \frac{2 y^{2}}{x (- 3 x + 4 y)} + \frac{6 x y}{x (- 3 x + 4 y)}$

Step 5.3.3.1.4

Cancel the common factor of $x$ .

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Step 5.3.3.1.4.1

Cancel the common factor.

$y' = - \frac{3 x}{- 3 x + 4 y} - \frac{2 y^{2}}{x (- 3 x + 4 y)} + \frac{6 x y}{x (- 3 x + 4 y)}$

Step 5.3.3.1.4.2

Rewrite the expression.

$y' = - \frac{3 x}{- 3 x + 4 y} - \frac{2 y^{2}}{x (- 3 x + 4 y)} + \frac{6 y}{- 3 x + 4 y}$

Step 5.3.3.2

To write $- \frac{3 x}{- 3 x + 4 y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$y' = - \frac{3 x}{- 3 x + 4 y} \cdot \frac{x}{x} - \frac{2 y^{2}}{x (- 3 x + 4 y)} + \frac{6 y}{- 3 x + 4 y}$

Step 5.3.3.3

Write each expression with a common denominator of $(- 3 x + 4 y) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.3.3.1

Multiply $\frac{3 x}{- 3 x + 4 y}$ by $\frac{x}{x}$ .

$y' = - \frac{3 x \cdot x}{(- 3 x + 4 y) x} - \frac{2 y^{2}}{x (- 3 x + 4 y)} + \frac{6 y}{- 3 x + 4 y}$

Step 5.3.3.3.2

Reorder the factors of $(- 3 x + 4 y) x$ .

$y' = - \frac{3 x \cdot x}{x (- 3 x + 4 y)} - \frac{2 y^{2}}{x (- 3 x + 4 y)} + \frac{6 y}{- 3 x + 4 y}$

Step 5.3.3.4

Combine the numerators over the common denominator.

$y' = \frac{- 3 x \cdot x - 2 y^{2}}{x (- 3 x + 4 y)} + \frac{6 y}{- 3 x + 4 y}$

Step 5.3.3.5

Multiply $x$ by $x$ by adding the exponents.

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Step 5.3.3.5.1

Move $x$ .

$y' = \frac{- 3 (x \cdot x) - 2 y^{2}}{x (- 3 x + 4 y)} + \frac{6 y}{- 3 x + 4 y}$

Step 5.3.3.5.2

Multiply $x$ by $x$ .

$y' = \frac{- 3 x^{2} - 2 y^{2}}{x (- 3 x + 4 y)} + \frac{6 y}{- 3 x + 4 y}$

Step 5.3.3.6

To write $\frac{6 y}{- 3 x + 4 y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$y' = \frac{- 3 x^{2} - 2 y^{2}}{x (- 3 x + 4 y)} + \frac{6 y}{- 3 x + 4 y} \cdot \frac{x}{x}$

Step 5.3.3.7

Write each expression with a common denominator of $x (- 3 x + 4 y)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.3.7.1

Multiply $\frac{6 y}{- 3 x + 4 y}$ by $\frac{x}{x}$ .

$y' = \frac{- 3 x^{2} - 2 y^{2}}{x (- 3 x + 4 y)} + \frac{6 y x}{(- 3 x + 4 y) x}$

Step 5.3.3.7.2

Reorder the factors of $(- 3 x + 4 y) x$ .

$y' = \frac{- 3 x^{2} - 2 y^{2}}{x (- 3 x + 4 y)} + \frac{6 y x}{x (- 3 x + 4 y)}$

Step 5.3.3.8

Combine the numerators over the common denominator.

$y' = \frac{- 3 x^{2} - 2 y^{2} + 6 y x}{x (- 3 x + 4 y)}$

Step 5.3.3.9

Factor $- 1$ out of $- 3 x^{2}$ .

$y' = \frac{- (3 x^{2}) - 2 y^{2} + 6 y x}{x (- 3 x + 4 y)}$

Step 5.3.3.10

Factor $- 1$ out of $- 2 y^{2}$ .

$y' = \frac{- (3 x^{2}) - (2 y^{2}) + 6 y x}{x (- 3 x + 4 y)}$

Step 5.3.3.11

Factor $- 1$ out of $- (3 x^{2}) - (2 y^{2})$ .

$y' = \frac{- (3 x^{2} + 2 y^{2}) + 6 y x}{x (- 3 x + 4 y)}$

Step 5.3.3.12

Factor $- 1$ out of $6 y x$ .

$y' = \frac{- (3 x^{2} + 2 y^{2}) - (- 6 y x)}{x (- 3 x + 4 y)}$

Step 5.3.3.13

Factor $- 1$ out of $- (3 x^{2} + 2 y^{2}) - (- 6 y x)$ .

$y' = \frac{- (3 x^{2} + 2 y^{2} - 6 y x)}{x (- 3 x + 4 y)}$

Step 5.3.3.14

Rewrite $- (3 x^{2} + 2 y^{2} - 6 y x)$ as $- 1 (3 x^{2} + 2 y^{2} - 6 y x)$ .

$y' = \frac{- 1 (3 x^{2} + 2 y^{2} - 6 y x)}{x (- 3 x + 4 y)}$

Step 5.3.3.15

Factor $- 1$ out of $- 3 x$ .

$y' = \frac{- 1 (3 x^{2} + 2 y^{2} - 6 y x)}{x (- (3 x) + 4 y)}$

Step 5.3.3.16

Factor $- 1$ out of $4 y$ .

$y' = \frac{- 1 (3 x^{2} + 2 y^{2} - 6 y x)}{x (- (3 x) - (- 4 y))}$

Step 5.3.3.17

Factor $- 1$ out of $- (3 x) - (- 4 y)$ .

$y' = \frac{- 1 (3 x^{2} + 2 y^{2} - 6 y x)}{x (- (3 x - 4 y))}$

Step 5.3.3.18

Rewrite $- (3 x - 4 y)$ as $- 1 (3 x - 4 y)$ .

$y' = \frac{- 1 (3 x^{2} + 2 y^{2} - 6 y x)}{x (- 1 (3 x - 4 y))}$

Step 5.3.3.19

Cancel the common factor.

$y' = \frac{- 1 (3 x^{2} + 2 y^{2} - 6 y x)}{x (- 1 (3 x - 4 y))}$

Step 5.3.3.20

Rewrite the expression.

$y' = \frac{3 x^{2} + 2 y^{2} - 6 y x}{x (3 x - 4 y)}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{3 x^{2} + 2 y^{2} - 6 y x}{x (3 x - 4 y)}$

Step 7

Set $\frac{d y}{d x} = 0$ then solve for $x$ in terms of $y$ .

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Step 7.1

Set the numerator equal to zero.

$3 x^{2} + 2 y^{2} - 6 y x = 0$

Step 7.2

Solve the equation for $x$ .

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Step 7.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7.2.2

Substitute the values $a = 3$ , $b = - 6 y$ , and $c = 2 y^{2}$ into the quadratic formula and solve for $x$ .

$\frac{6 y \pm \sqrt{{(- 6 y)}^{2} - 4 \cdot (3 \cdot (2 y^{2}))}}{2 \cdot 3}$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Simplify the numerator.

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Step 7.2.3.1.1

Add parentheses.

$x = \frac{6 y \pm \sqrt{{(- 6 y)}^{2} - 4 \cdot (3 \cdot (2 y^{2}))}}{2 \cdot 3}$

Step 7.2.3.1.2

Let $u = 3 \cdot (2 y^{2})$ . Substitute $u$ for all occurrences of $3 \cdot (2 y^{2})$ .

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Step 7.2.3.1.2.1

Apply the product rule to $- 6 y$ .

$x = \frac{6 y \pm \sqrt{{(- 6)}^{2} y^{2} - 4 \cdot u}}{2 \cdot 3}$

Step 7.2.3.1.2.2

Raise $- 6$ to the power of $2$ .

$x = \frac{6 y \pm \sqrt{36 y^{2} - 4 u}}{2 \cdot 3}$

Step 7.2.3.1.3

Factor $4$ out of $36 y^{2} - 4 u$ .

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Step 7.2.3.1.3.1

Factor $4$ out of $36 y^{2}$ .

$x = \frac{6 y \pm \sqrt{4 (9 y^{2}) - 4 u}}{2 \cdot 3}$

Step 7.2.3.1.3.2

Factor $4$ out of $- 4 u$ .

$x = \frac{6 y \pm \sqrt{4 (9 y^{2}) + 4 (- u)}}{2 \cdot 3}$

Step 7.2.3.1.3.3

Factor $4$ out of $4 (9 y^{2}) + 4 (- u)$ .

$x = \frac{6 y \pm \sqrt{4 (9 y^{2} - u)}}{2 \cdot 3}$

Step 7.2.3.1.4

Replace all occurrences of $u$ with $3 \cdot (2 y^{2})$ .

$x = \frac{6 y \pm \sqrt{4 (9 y^{2} - (3 \cdot (2 y^{2})))}}{2 \cdot 3}$

Step 7.2.3.1.5

Simplify.

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Step 7.2.3.1.5.1

Simplify each term.

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Step 7.2.3.1.5.1.1

Multiply $2$ by $3$ .

$x = \frac{6 y \pm \sqrt{4 (9 y^{2} - (6 y^{2}))}}{2 \cdot 3}$

Step 7.2.3.1.5.1.2

Multiply $6$ by $- 1$ .

$x = \frac{6 y \pm \sqrt{4 (9 y^{2} - 6 y^{2})}}{2 \cdot 3}$

Step 7.2.3.1.5.2

Subtract $6 y^{2}$ from $9 y^{2}$ .

$x = \frac{6 y \pm \sqrt{4 (3 y^{2})}}{2 \cdot 3}$

$x = \frac{6 y \pm \sqrt{4 \cdot (3 y^{2})}}{2 \cdot 3}$

Step 7.2.3.1.6

Multiply $4$ by $3$ .

$x = \frac{6 y \pm \sqrt{12 y^{2}}}{2 \cdot 3}$

Step 7.2.3.1.7

Rewrite $12 y^{2}$ as ${(2 y)}^{2} \cdot 3$ .

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Step 7.2.3.1.7.1

Factor $4$ out of $12$ .

$x = \frac{6 y \pm \sqrt{4 (3) y^{2}}}{2 \cdot 3}$

Step 7.2.3.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{6 y \pm \sqrt{2^{2} \cdot (3 y^{2})}}{2 \cdot 3}$

Step 7.2.3.1.7.3

Move $3$ .

$x = \frac{6 y \pm \sqrt{2^{2} y^{2} \cdot 3}}{2 \cdot 3}$

Step 7.2.3.1.7.4

Rewrite $2^{2} y^{2}$ as ${(2 y)}^{2}$ .

$x = \frac{6 y \pm \sqrt{{(2 y)}^{2} \cdot 3}}{2 \cdot 3}$

Step 7.2.3.1.8

Pull terms out from under the radical.

$x = \frac{6 y \pm 2 y \sqrt{3}}{2 \cdot 3}$

Step 7.2.3.2

Multiply $2$ by $3$ .

$x = \frac{6 y \pm 2 y \sqrt{3}}{6}$

Step 7.2.3.3

Simplify $\frac{6 y \pm 2 y \sqrt{3}}{6}$ .

$x = \frac{3 y \pm y \sqrt{3}}{3}$

Step 7.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 7.2.4.1

Simplify the numerator.

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Step 7.2.4.1.1

Add parentheses.

$x = \frac{6 y \pm \sqrt{{(- 6 y)}^{2} - 4 \cdot (3 \cdot (2 y^{2}))}}{2 \cdot 3}$

Step 7.2.4.1.2

Let $u = 3 \cdot (2 y^{2})$ . Substitute $u$ for all occurrences of $3 \cdot (2 y^{2})$ .

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Step 7.2.4.1.2.1

Apply the product rule to $- 6 y$ .

$x = \frac{6 y \pm \sqrt{{(- 6)}^{2} y^{2} - 4 \cdot u}}{2 \cdot 3}$

Step 7.2.4.1.2.2

Raise $- 6$ to the power of $2$ .

$x = \frac{6 y \pm \sqrt{36 y^{2} - 4 u}}{2 \cdot 3}$

Step 7.2.4.1.3

Factor $4$ out of $36 y^{2} - 4 u$ .

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Step 7.2.4.1.3.1

Factor $4$ out of $36 y^{2}$ .

$x = \frac{6 y \pm \sqrt{4 (9 y^{2}) - 4 u}}{2 \cdot 3}$

Step 7.2.4.1.3.2

Factor $4$ out of $- 4 u$ .

$x = \frac{6 y \pm \sqrt{4 (9 y^{2}) + 4 (- u)}}{2 \cdot 3}$

Step 7.2.4.1.3.3

Factor $4$ out of $4 (9 y^{2}) + 4 (- u)$ .

$x = \frac{6 y \pm \sqrt{4 (9 y^{2} - u)}}{2 \cdot 3}$

Step 7.2.4.1.4

Replace all occurrences of $u$ with $3 \cdot (2 y^{2})$ .

$x = \frac{6 y \pm \sqrt{4 (9 y^{2} - (3 \cdot (2 y^{2})))}}{2 \cdot 3}$

Step 7.2.4.1.5

Simplify.

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Step 7.2.4.1.5.1

Simplify each term.

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Step 7.2.4.1.5.1.1

Multiply $2$ by $3$ .

$x = \frac{6 y \pm \sqrt{4 (9 y^{2} - (6 y^{2}))}}{2 \cdot 3}$

Step 7.2.4.1.5.1.2

Multiply $6$ by $- 1$ .

$x = \frac{6 y \pm \sqrt{4 (9 y^{2} - 6 y^{2})}}{2 \cdot 3}$

Step 7.2.4.1.5.2

Subtract $6 y^{2}$ from $9 y^{2}$ .

$x = \frac{6 y \pm \sqrt{4 (3 y^{2})}}{2 \cdot 3}$

$x = \frac{6 y \pm \sqrt{4 \cdot (3 y^{2})}}{2 \cdot 3}$

Step 7.2.4.1.6

Multiply $4$ by $3$ .

$x = \frac{6 y \pm \sqrt{12 y^{2}}}{2 \cdot 3}$

Step 7.2.4.1.7

Rewrite $12 y^{2}$ as ${(2 y)}^{2} \cdot 3$ .

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Step 7.2.4.1.7.1

Factor $4$ out of $12$ .

$x = \frac{6 y \pm \sqrt{4 (3) y^{2}}}{2 \cdot 3}$

Step 7.2.4.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{6 y \pm \sqrt{2^{2} \cdot (3 y^{2})}}{2 \cdot 3}$

Step 7.2.4.1.7.3

Move $3$ .

$x = \frac{6 y \pm \sqrt{2^{2} y^{2} \cdot 3}}{2 \cdot 3}$

Step 7.2.4.1.7.4

Rewrite $2^{2} y^{2}$ as ${(2 y)}^{2}$ .

$x = \frac{6 y \pm \sqrt{{(2 y)}^{2} \cdot 3}}{2 \cdot 3}$

Step 7.2.4.1.8

Pull terms out from under the radical.

$x = \frac{6 y \pm 2 y \sqrt{3}}{2 \cdot 3}$

Step 7.2.4.2

Multiply $2$ by $3$ .

$x = \frac{6 y \pm 2 y \sqrt{3}}{6}$

Step 7.2.4.3

Simplify $\frac{6 y \pm 2 y \sqrt{3}}{6}$ .

$x = \frac{3 y \pm y \sqrt{3}}{3}$

Step 7.2.4.4

Change the $\pm$ to $+$ .

$x = \frac{3 y + y \sqrt{3}}{3}$

Step 7.2.4.5

Factor $y$ out of $3 y + y \sqrt{3}$ .

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Step 7.2.4.5.1

Factor $y$ out of $3 y$ .

$x = \frac{y \cdot 3 + y \sqrt{3}}{3}$

Step 7.2.4.5.2

Factor $y$ out of $y \sqrt{3}$ .

$x = \frac{y \cdot 3 + y (\sqrt{3})}{3}$

Step 7.2.4.5.3

Factor $y$ out of $y \cdot 3 + y (\sqrt{3})$ .

$x = \frac{y (3 + \sqrt{3})}{3}$

Step 7.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.2.5.1

Simplify the numerator.

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Step 7.2.5.1.1

Add parentheses.

$x = \frac{6 y \pm \sqrt{{(- 6 y)}^{2} - 4 \cdot (3 \cdot (2 y^{2}))}}{2 \cdot 3}$

Step 7.2.5.1.2

Let $u = 3 \cdot (2 y^{2})$ . Substitute $u$ for all occurrences of $3 \cdot (2 y^{2})$ .

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Step 7.2.5.1.2.1

Apply the product rule to $- 6 y$ .

$x = \frac{6 y \pm \sqrt{{(- 6)}^{2} y^{2} - 4 \cdot u}}{2 \cdot 3}$

Step 7.2.5.1.2.2

Raise $- 6$ to the power of $2$ .

$x = \frac{6 y \pm \sqrt{36 y^{2} - 4 u}}{2 \cdot 3}$

Step 7.2.5.1.3

Factor $4$ out of $36 y^{2} - 4 u$ .

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Step 7.2.5.1.3.1

Factor $4$ out of $36 y^{2}$ .

$x = \frac{6 y \pm \sqrt{4 (9 y^{2}) - 4 u}}{2 \cdot 3}$

Step 7.2.5.1.3.2

Factor $4$ out of $- 4 u$ .

$x = \frac{6 y \pm \sqrt{4 (9 y^{2}) + 4 (- u)}}{2 \cdot 3}$

Step 7.2.5.1.3.3

Factor $4$ out of $4 (9 y^{2}) + 4 (- u)$ .

$x = \frac{6 y \pm \sqrt{4 (9 y^{2} - u)}}{2 \cdot 3}$

Step 7.2.5.1.4

Replace all occurrences of $u$ with $3 \cdot (2 y^{2})$ .

$x = \frac{6 y \pm \sqrt{4 (9 y^{2} - (3 \cdot (2 y^{2})))}}{2 \cdot 3}$

Step 7.2.5.1.5

Simplify.

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Step 7.2.5.1.5.1

Simplify each term.

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Step 7.2.5.1.5.1.1

Multiply $2$ by $3$ .

$x = \frac{6 y \pm \sqrt{4 (9 y^{2} - (6 y^{2}))}}{2 \cdot 3}$

Step 7.2.5.1.5.1.2

Multiply $6$ by $- 1$ .

$x = \frac{6 y \pm \sqrt{4 (9 y^{2} - 6 y^{2})}}{2 \cdot 3}$

Step 7.2.5.1.5.2

Subtract $6 y^{2}$ from $9 y^{2}$ .

$x = \frac{6 y \pm \sqrt{4 (3 y^{2})}}{2 \cdot 3}$

$x = \frac{6 y \pm \sqrt{4 \cdot (3 y^{2})}}{2 \cdot 3}$

Step 7.2.5.1.6

Multiply $4$ by $3$ .

$x = \frac{6 y \pm \sqrt{12 y^{2}}}{2 \cdot 3}$

Step 7.2.5.1.7

Rewrite $12 y^{2}$ as ${(2 y)}^{2} \cdot 3$ .

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Step 7.2.5.1.7.1

Factor $4$ out of $12$ .

$x = \frac{6 y \pm \sqrt{4 (3) y^{2}}}{2 \cdot 3}$

Step 7.2.5.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{6 y \pm \sqrt{2^{2} \cdot (3 y^{2})}}{2 \cdot 3}$

Step 7.2.5.1.7.3

Move $3$ .

$x = \frac{6 y \pm \sqrt{2^{2} y^{2} \cdot 3}}{2 \cdot 3}$

Step 7.2.5.1.7.4

Rewrite $2^{2} y^{2}$ as ${(2 y)}^{2}$ .

$x = \frac{6 y \pm \sqrt{{(2 y)}^{2} \cdot 3}}{2 \cdot 3}$

Step 7.2.5.1.8

Pull terms out from under the radical.

$x = \frac{6 y \pm 2 y \sqrt{3}}{2 \cdot 3}$

Step 7.2.5.2

Multiply $2$ by $3$ .

$x = \frac{6 y \pm 2 y \sqrt{3}}{6}$

Step 7.2.5.3

Simplify $\frac{6 y \pm 2 y \sqrt{3}}{6}$ .

$x = \frac{3 y \pm y \sqrt{3}}{3}$

Step 7.2.5.4

Change the $\pm$ to $-$ .

$x = \frac{3 y - y \sqrt{3}}{3}$

Step 7.2.5.5

Simplify the numerator.

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Step 7.2.5.5.1

Factor $y$ out of $3 y - y \sqrt{3}$ .

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Step 7.2.5.5.1.1

Factor $y$ out of $3 y$ .

$x = \frac{y \cdot 3 - y \sqrt{3}}{3}$

Step 7.2.5.5.1.2

Factor $y$ out of $- y \sqrt{3}$ .

$x = \frac{y \cdot 3 + y (- 1 \sqrt{3})}{3}$

Step 7.2.5.5.1.3

Factor $y$ out of $y \cdot 3 + y (- 1 \sqrt{3})$ .

$x = \frac{y (3 - 1 \sqrt{3})}{3}$

Step 7.2.5.5.2

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

$x = \frac{y (3 - \sqrt{3})}{3}$

Step 7.2.6

The final answer is the combination of both solutions.

$x = \frac{y (3 + \sqrt{3})}{3}$

$x = \frac{y (3 - \sqrt{3})}{3}$

$x = \frac{y (3 + \sqrt{3})}{3}$

$x = \frac{y (3 - \sqrt{3})}{3}$

$x = \frac{y (3 + \sqrt{3})}{3}$

$x = \frac{y (3 - \sqrt{3})}{3}$

Step 8

Solve for $y$ .

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Step 8.1

Simplify ${(\frac{y (3 + \sqrt{3})}{3})}^{3} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2}$ .

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Step 8.1.1

Simplify each term.

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Step 8.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 8.1.1.1.1

Apply the product rule to $\frac{y (3 + \sqrt{3})}{3}$ .

$\frac{{(y (3 + \sqrt{3}))}^{3}}{3^{3}} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.1.2

Apply the product rule to $y (3 + \sqrt{3})$ .

$\frac{y^{3} {(3 + \sqrt{3})}^{3}}{3^{3}} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.2

Raise $3$ to the power of $3$ .

$\frac{y^{3} {(3 + \sqrt{3})}^{3}}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.3

Use the Binomial Theorem.

$\frac{y^{3} (3^{3} + 3 \cdot 3^{2} \sqrt{3} + 3 \cdot 3 {\sqrt{3}}^{2} + {\sqrt{3}}^{3})}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.4

Simplify each term.

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Step 8.1.1.4.1

Raise $3$ to the power of $3$ .

$\frac{y^{3} (27 + 3 \cdot 3^{2} \sqrt{3} + 3 \cdot 3 {\sqrt{3}}^{2} + {\sqrt{3}}^{3})}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.4.2

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 8.1.1.4.2.1

Multiply $3$ by $3^{2}$ .

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Step 8.1.1.4.2.1.1

Raise $3$ to the power of $1$ .

$\frac{y^{3} (27 + 3^{1} \cdot 3^{2} \sqrt{3} + 3 \cdot 3 {\sqrt{3}}^{2} + {\sqrt{3}}^{3})}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.4.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{y^{3} (27 + 3^{1 + 2} \sqrt{3} + 3 \cdot 3 {\sqrt{3}}^{2} + {\sqrt{3}}^{3})}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.4.2.2

Add $1$ and $2$ .

$\frac{y^{3} (27 + 3^{3} \sqrt{3} + 3 \cdot 3 {\sqrt{3}}^{2} + {\sqrt{3}}^{3})}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.4.3

Raise $3$ to the power of $3$ .

$\frac{y^{3} (27 + 27 \sqrt{3} + 3 \cdot 3 {\sqrt{3}}^{2} + {\sqrt{3}}^{3})}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.4.4

Multiply $3$ by $3$ .

$\frac{y^{3} (27 + 27 \sqrt{3} + 9 {\sqrt{3}}^{2} + {\sqrt{3}}^{3})}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.4.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 8.1.1.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{y^{3} (27 + 27 \sqrt{3} + 9 {(3^{\frac{1}{2}})}^{2} + {\sqrt{3}}^{3})}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.4.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{y^{3} (27 + 27 \sqrt{3} + 9 \cdot 3^{\frac{1}{2} \cdot 2} + {\sqrt{3}}^{3})}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.4.5.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{y^{3} (27 + 27 \sqrt{3} + 9 \cdot 3^{\frac{2}{2}} + {\sqrt{3}}^{3})}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.4.5.4

Cancel the common factor of $2$ .

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Step 8.1.1.4.5.4.1

Cancel the common factor.

$\frac{y^{3} (27 + 27 \sqrt{3} + 9 \cdot 3^{\frac{2}{2}} + {\sqrt{3}}^{3})}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.4.5.4.2

Rewrite the expression.

$\frac{y^{3} (27 + 27 \sqrt{3} + 9 \cdot 3^{1} + {\sqrt{3}}^{3})}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.4.5.5

Evaluate the exponent.

$\frac{y^{3} (27 + 27 \sqrt{3} + 9 \cdot 3 + {\sqrt{3}}^{3})}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.4.6

Multiply $9$ by $3$ .

$\frac{y^{3} (27 + 27 \sqrt{3} + 27 + {\sqrt{3}}^{3})}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.4.7

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$\frac{y^{3} (27 + 27 \sqrt{3} + 27 + \sqrt{3^{3}})}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.4.8

Raise $3$ to the power of $3$ .

$\frac{y^{3} (27 + 27 \sqrt{3} + 27 + \sqrt{27})}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.4.9

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 8.1.1.4.9.1

Factor $9$ out of $27$ .

$\frac{y^{3} (27 + 27 \sqrt{3} + 27 + \sqrt{9 (3)})}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.4.9.2

Rewrite $9$ as $3^{2}$ .

$\frac{y^{3} (27 + 27 \sqrt{3} + 27 + \sqrt{3^{2} \cdot 3})}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.4.10

Pull terms out from under the radical.

$\frac{y^{3} (27 + 27 \sqrt{3} + 27 + 3 \sqrt{3})}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.5

Add $27$ and $27$ .

$\frac{y^{3} (54 + 27 \sqrt{3} + 3 \sqrt{3})}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.6

Add $27 \sqrt{3}$ and $3 \sqrt{3}$ .

$\frac{y^{3} (54 + 30 \sqrt{3})}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.7

Cancel the common factor of $54 + 30 \sqrt{3}$ and $27$ .

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Step 8.1.1.7.1

Factor $3$ out of $y^{3} (54 + 30 \sqrt{3})$ .

$\frac{3 (y^{3} (18 + 10 \sqrt{3}))}{27} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.7.2

Cancel the common factors.

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Step 8.1.1.7.2.1

Factor $3$ out of $27$ .

$\frac{3 (y^{3} (18 + 10 \sqrt{3}))}{3 (9)} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.7.2.2

Cancel the common factor.

$\frac{3 (y^{3} (18 + 10 \sqrt{3}))}{3 \cdot 9} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.7.2.3

Rewrite the expression.

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 3 {(\frac{y (3 + \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.8

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 8.1.1.8.1

Apply the product rule to $\frac{y (3 + \sqrt{3})}{3}$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 3 \frac{{(y (3 + \sqrt{3}))}^{2}}{3^{2}} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.8.2

Apply the product rule to $y (3 + \sqrt{3})$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 3 \frac{y^{2} {(3 + \sqrt{3})}^{2}}{3^{2}} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.9

Raise $3$ to the power of $2$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 3 \frac{y^{2} {(3 + \sqrt{3})}^{2}}{9} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.10

Cancel the common factor of $3$ .

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Step 8.1.1.10.1

Factor $3$ out of $- 3$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} + 3 (- 1) \frac{y^{2} {(3 + \sqrt{3})}^{2}}{9} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.10.2

Factor $3$ out of $9$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} + 3 \cdot - 1 \frac{y^{2} {(3 + \sqrt{3})}^{2}}{3 \cdot 3} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.10.3

Cancel the common factor.

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} + 3 \cdot - 1 \frac{y^{2} {(3 + \sqrt{3})}^{2}}{3 \cdot 3} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.10.4

Rewrite the expression.

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 \frac{y^{2} {(3 + \sqrt{3})}^{2}}{3} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.11

Rewrite ${(3 + \sqrt{3})}^{2}$ as $(3 + \sqrt{3}) (3 + \sqrt{3})$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 \frac{y^{2} ((3 + \sqrt{3}) (3 + \sqrt{3}))}{3} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.12

Expand $(3 + \sqrt{3}) (3 + \sqrt{3})$ using the FOIL Method.

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Step 8.1.1.12.1

Apply the distributive property.

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 \frac{y^{2} (3 (3 + \sqrt{3}) + \sqrt{3} (3 + \sqrt{3}))}{3} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.12.2

Apply the distributive property.

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 \frac{y^{2} (3 \cdot 3 + 3 \sqrt{3} + \sqrt{3} (3 + \sqrt{3}))}{3} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.12.3

Apply the distributive property.

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 \frac{y^{2} (3 \cdot 3 + 3 \sqrt{3} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3})}{3} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.13

Simplify and combine like terms.

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Step 8.1.1.13.1

Simplify each term.

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Step 8.1.1.13.1.1

Multiply $3$ by $3$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 \frac{y^{2} (9 + 3 \sqrt{3} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3})}{3} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.13.1.2

Move $3$ to the left of $\sqrt{3}$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 \frac{y^{2} (9 + 3 \sqrt{3} + 3 \cdot \sqrt{3} + \sqrt{3} \sqrt{3})}{3} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.13.1.3

Combine using the product rule for radicals.

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 \frac{y^{2} (9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{3 \cdot 3})}{3} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.13.1.4

Multiply $3$ by $3$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 \frac{y^{2} (9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{9})}{3} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.13.1.5

Rewrite $9$ as $3^{2}$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 \frac{y^{2} (9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{3^{2}})}{3} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.13.1.6

Pull terms out from under the radical, assuming positive real numbers.

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 \frac{y^{2} (9 + 3 \sqrt{3} + 3 \sqrt{3} + 3)}{3} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.13.2

Add $9$ and $3$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 \frac{y^{2} (12 + 3 \sqrt{3} + 3 \sqrt{3})}{3} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.13.3

Add $3 \sqrt{3}$ and $3 \sqrt{3}$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 \frac{y^{2} (12 + 6 \sqrt{3})}{3} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.14

Cancel the common factor of $12 + 6 \sqrt{3}$ and $3$ .

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Step 8.1.1.14.1

Factor $3$ out of $y^{2} (12 + 6 \sqrt{3})$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 \frac{3 (y^{2} (4 + 2 \sqrt{3}))}{3} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.14.2

Cancel the common factors.

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Step 8.1.1.14.2.1

Factor $3$ out of $3$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 \frac{3 (y^{2} (4 + 2 \sqrt{3}))}{3 (1)} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.14.2.2

Cancel the common factor.

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 \frac{3 (y^{2} (4 + 2 \sqrt{3}))}{3 \cdot 1} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.14.2.3

Rewrite the expression.

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 \frac{y^{2} (4 + 2 \sqrt{3})}{1} y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.14.2.4

Divide $y^{2} (4 + 2 \sqrt{3})$ by $1$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 (y^{2} (4 + 2 \sqrt{3})) y + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.15

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 8.1.1.15.1

Move $y$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 (y \cdot y^{2} (4 + 2 \sqrt{3})) + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.15.2

Multiply $y$ by $y^{2}$ .

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Step 8.1.1.15.2.1

Raise $y$ to the power of $1$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 (y^{1} y^{2} (4 + 2 \sqrt{3})) + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.15.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 (y^{1 + 2} (4 + 2 \sqrt{3})) + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.15.3

Add $1$ and $2$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 (y^{3} (4 + 2 \sqrt{3})) + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.16

Apply the distributive property.

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 (y^{3} \cdot 4 + y^{3} (2 \sqrt{3})) + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.17

Move $4$ to the left of $y^{3}$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 (4 \cdot y^{3} + y^{3} (2 \sqrt{3})) + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.18

Move $2$ to the left of $y^{3}$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 (4 y^{3} + 2 y^{3} \sqrt{3}) + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.19

Apply the distributive property.

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 1 (4 y^{3}) - 1 (2 y^{3} \sqrt{3}) + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.20

Multiply $4$ by $- 1$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 4 y^{3} - 1 (2 y^{3} \sqrt{3}) + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.21

Multiply $2$ by $- 1$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 4 y^{3} - 2 (y^{3} \sqrt{3}) + 2 (\frac{y (3 + \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 8.1.1.22

Combine $2$ and $\frac{y (3 + \sqrt{3})}{3}$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 4 y^{3} - 2 y^{3} \sqrt{3} + \frac{2 (y (3 + \sqrt{3}))}{3} \cdot y^{2} = 12$

Step 8.1.1.23

Multiply $\frac{2 y (3 + \sqrt{3})}{3} y^{2}$ .

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Step 8.1.1.23.1

Combine $\frac{2 y (3 + \sqrt{3})}{3}$ and $y^{2}$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 4 y^{3} - 2 y^{3} \sqrt{3} + \frac{2 y (3 + \sqrt{3}) y^{2}}{3} = 12$

Step 8.1.1.23.2

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 8.1.1.23.2.1

Move $y^{2}$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 4 y^{3} - 2 y^{3} \sqrt{3} + \frac{2 (y^{2} y) (3 + \sqrt{3})}{3} = 12$

Step 8.1.1.23.2.2

Multiply $y^{2}$ by $y$ .

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Step 8.1.1.23.2.2.1

Raise $y$ to the power of $1$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 4 y^{3} - 2 y^{3} \sqrt{3} + \frac{2 (y^{2} y^{1}) (3 + \sqrt{3})}{3} = 12$

Step 8.1.1.23.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 4 y^{3} - 2 y^{3} \sqrt{3} + \frac{2 y^{2 + 1} (3 + \sqrt{3})}{3} = 12$

Step 8.1.1.23.2.3

Add $2$ and $1$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 4 y^{3} - 2 y^{3} \sqrt{3} + \frac{2 y^{3} (3 + \sqrt{3})}{3} = 12$

Step 8.1.2

To write $- 4 y^{3}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} - 4 y^{3} \cdot \frac{9}{9} - 2 y^{3} \sqrt{3} + \frac{2 y^{3} (3 + \sqrt{3})}{3} = 12$

Step 8.1.3

Simplify terms.

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Step 8.1.3.1

Combine $- 4 y^{3}$ and $\frac{9}{9}$ .

$\frac{y^{3} (18 + 10 \sqrt{3})}{9} + \frac{- 4 y^{3} \cdot 9}{9} - 2 y^{3} \sqrt{3} + \frac{2 y^{3} (3 + \sqrt{3})}{3} = 12$

Step 8.1.3.2

Combine the numerators over the common denominator.

$\frac{y^{3} (18 + 10 \sqrt{3}) - 4 y^{3} \cdot 9}{9} - 2 y^{3} \sqrt{3} + \frac{2 y^{3} (3 + \sqrt{3})}{3} = 12$

Step 8.1.4

Simplify the numerator.

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Step 8.1.4.1

Factor $y^{3}$ out of $y^{3} (18 + 10 \sqrt{3}) - 4 y^{3} \cdot 9$ .

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Step 8.1.4.1.1

Factor $y^{3}$ out of $- 4 y^{3} \cdot 9$ .

$\frac{y^{3} (18 + 10 \sqrt{3}) + y^{3} (- 4 \cdot 9)}{9} - 2 y^{3} \sqrt{3} + \frac{2 y^{3} (3 + \sqrt{3})}{3} = 12$

Step 8.1.4.1.2

Factor $y^{3}$ out of $y^{3} (18 + 10 \sqrt{3}) + y^{3} (- 4 \cdot 9)$ .

$\frac{y^{3} (18 + 10 \sqrt{3} - 4 \cdot 9)}{9} - 2 y^{3} \sqrt{3} + \frac{2 y^{3} (3 + \sqrt{3})}{3} = 12$

Step 8.1.4.2

Multiply $- 4$ by $9$ .

$\frac{y^{3} (18 + 10 \sqrt{3} - 36)}{9} - 2 y^{3} \sqrt{3} + \frac{2 y^{3} (3 + \sqrt{3})}{3} = 12$

Step 8.1.4.3

Subtract $36$ from $18$ .

$\frac{y^{3} (- 18 + 10 \sqrt{3})}{9} - 2 y^{3} \sqrt{3} + \frac{2 y^{3} (3 + \sqrt{3})}{3} = 12$

Step 8.1.5

Find the common denominator.

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Step 8.1.5.1

Write $- 2 y^{3} \sqrt{3}$ as a fraction with denominator $1$ .

$\frac{y^{3} (- 18 + 10 \sqrt{3})}{9} + \frac{- 2 y^{3} \sqrt{3}}{1} + \frac{2 y^{3} (3 + \sqrt{3})}{3} = 12$

Step 8.1.5.2

Multiply $\frac{- 2 y^{3} \sqrt{3}}{1}$ by $\frac{9}{9}$ .

$\frac{y^{3} (- 18 + 10 \sqrt{3})}{9} + \frac{- 2 y^{3} \sqrt{3}}{1} \cdot \frac{9}{9} + \frac{2 y^{3} (3 + \sqrt{3})}{3} = 12$

Step 8.1.5.3

Multiply $\frac{- 2 y^{3} \sqrt{3}}{1}$ by $\frac{9}{9}$ .

$\frac{y^{3} (- 18 + 10 \sqrt{3})}{9} + \frac{- 2 y^{3} \sqrt{3} \cdot 9}{9} + \frac{2 y^{3} (3 + \sqrt{3})}{3} = 12$

Step 8.1.5.4

Multiply $\frac{2 y^{3} (3 + \sqrt{3})}{3}$ by $\frac{3}{3}$ .

$\frac{y^{3} (- 18 + 10 \sqrt{3})}{9} + \frac{- 2 y^{3} \sqrt{3} \cdot 9}{9} + \frac{2 y^{3} (3 + \sqrt{3})}{3} \cdot \frac{3}{3} = 12$

Step 8.1.5.5

Multiply $\frac{2 y^{3} (3 + \sqrt{3})}{3}$ by $\frac{3}{3}$ .

$\frac{y^{3} (- 18 + 10 \sqrt{3})}{9} + \frac{- 2 y^{3} \sqrt{3} \cdot 9}{9} + \frac{2 y^{3} (3 + \sqrt{3}) \cdot 3}{3 \cdot 3} = 12$

Step 8.1.5.6

Multiply $3$ by $3$ .

$\frac{y^{3} (- 18 + 10 \sqrt{3})}{9} + \frac{- 2 y^{3} \sqrt{3} \cdot 9}{9} + \frac{2 y^{3} (3 + \sqrt{3}) \cdot 3}{9} = 12$

Step 8.1.6

Combine the numerators over the common denominator.

$\frac{y^{3} (- 18 + 10 \sqrt{3}) - 2 y^{3} \sqrt{3} \cdot 9 + 2 y^{3} (3 + \sqrt{3}) \cdot 3}{9} = 12$

Step 8.1.7

Simplify each term.

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Step 8.1.7.1

Apply the distributive property.

$\frac{y^{3} \cdot - 18 + y^{3} (10 \sqrt{3}) - 2 y^{3} \sqrt{3} \cdot 9 + 2 y^{3} (3 + \sqrt{3}) \cdot 3}{9} = 12$

Step 8.1.7.2

Move $- 18$ to the left of $y^{3}$ .

$\frac{- 18 \cdot y^{3} + y^{3} (10 \sqrt{3}) - 2 y^{3} \sqrt{3} \cdot 9 + 2 y^{3} (3 + \sqrt{3}) \cdot 3}{9} = 12$

Step 8.1.7.3

Move $10$ to the left of $y^{3}$ .

$\frac{- 18 y^{3} + 10 y^{3} \sqrt{3} - 2 y^{3} \sqrt{3} \cdot 9 + 2 y^{3} (3 + \sqrt{3}) \cdot 3}{9} = 12$

Step 8.1.7.4

Multiply $9$ by $- 2$ .

$\frac{- 18 y^{3} + 10 y^{3} \sqrt{3} - 18 y^{3} \sqrt{3} + 2 y^{3} (3 + \sqrt{3}) \cdot 3}{9} = 12$

Step 8.1.7.5

Apply the distributive property.

$\frac{- 18 y^{3} + 10 y^{3} \sqrt{3} - 18 y^{3} \sqrt{3} + (2 y^{3} \cdot 3 + 2 y^{3} \sqrt{3}) \cdot 3}{9} = 12$

Step 8.1.7.6

Multiply $3$ by $2$ .

$\frac{- 18 y^{3} + 10 y^{3} \sqrt{3} - 18 y^{3} \sqrt{3} + (6 y^{3} + 2 y^{3} \sqrt{3}) \cdot 3}{9} = 12$

Step 8.1.7.7

Apply the distributive property.

$\frac{- 18 y^{3} + 10 y^{3} \sqrt{3} - 18 y^{3} \sqrt{3} + 6 y^{3} \cdot 3 + 2 y^{3} \sqrt{3} \cdot 3}{9} = 12$

Step 8.1.7.8

Multiply $3$ by $6$ .

$\frac{- 18 y^{3} + 10 y^{3} \sqrt{3} - 18 y^{3} \sqrt{3} + 18 y^{3} + 2 y^{3} \sqrt{3} \cdot 3}{9} = 12$

Step 8.1.7.9

Multiply $3$ by $2$ .

$\frac{- 18 y^{3} + 10 y^{3} \sqrt{3} - 18 y^{3} \sqrt{3} + 18 y^{3} + 6 y^{3} \sqrt{3}}{9} = 12$

Step 8.1.8

Simplify by adding terms.

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Step 8.1.8.1

Combine the opposite terms in $- 18 y^{3} + 10 y^{3} \sqrt{3} - 18 y^{3} \sqrt{3} + 18 y^{3} + 6 y^{3} \sqrt{3}$ .

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Step 8.1.8.1.1

Add $- 18 y^{3}$ and $18 y^{3}$ .

$\frac{10 y^{3} \sqrt{3} - 18 y^{3} \sqrt{3} + 0 + 6 y^{3} \sqrt{3}}{9} = 12$

Step 8.1.8.1.2

Add $10 y^{3} \sqrt{3} - 18 y^{3} \sqrt{3}$ and $0$ .

$\frac{10 y^{3} \sqrt{3} - 18 y^{3} \sqrt{3} + 6 y^{3} \sqrt{3}}{9} = 12$

Step 8.1.8.2

Subtract $18 y^{3} \sqrt{3}$ from $10 y^{3} \sqrt{3}$ .

$\frac{- 8 y^{3} \sqrt{3} + 6 y^{3} \sqrt{3}}{9} = 12$

Step 8.1.8.3

Add $- 8 y^{3} \sqrt{3}$ and $6 y^{3} \sqrt{3}$ .

$\frac{- 2 y^{3} \sqrt{3}}{9} = 12$

Step 8.1.8.4

Move the negative in front of the fraction.

$- \frac{2 y^{3} \sqrt{3}}{9} = 12$

Step 8.2

Multiply both sides of the equation by $\frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}}$ .

$\frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}} (- \frac{2 y^{3} \sqrt{3}}{9}) = \frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}} \cdot 12$

Step 8.3

Simplify both sides of the equation.

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Step 8.3.1

Simplify the left side.

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Step 8.3.1.1

Simplify $\frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}} (- \frac{2 y^{3} \sqrt{3}}{9})$ .

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Step 8.3.1.1.1

Cancel the common factor of $2$ .

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Step 8.3.1.1.1.1

Move the leading negative in $- \frac{2 y^{3} \sqrt{3}}{9}$ into the numerator.

$\frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}} \cdot \frac{- 2 y^{3} \sqrt{3}}{9} = \frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}} \cdot 12$

Step 8.3.1.1.1.2

Factor $2$ out of $2 \cdot - 1 \frac{\sqrt{3}}{9}$ .

$\frac{1}{2 (- \frac{\sqrt{3}}{9})} \cdot \frac{- 2 y^{3} \sqrt{3}}{9} = \frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}} \cdot 12$

Step 8.3.1.1.1.3

Factor $2$ out of $- 2 y^{3} \sqrt{3}$ .

$\frac{1}{2 (- \frac{\sqrt{3}}{9})} \cdot \frac{2 (- y^{3} \sqrt{3})}{9} = \frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}} \cdot 12$

Step 8.3.1.1.1.4

Cancel the common factor.

$\frac{1}{2 (- \frac{\sqrt{3}}{9})} \cdot \frac{2 (- y^{3} \sqrt{3})}{9} = \frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}} \cdot 12$

Step 8.3.1.1.1.5

Rewrite the expression.

$\frac{1}{- \frac{\sqrt{3}}{9}} \cdot \frac{- y^{3} \sqrt{3}}{9} = \frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}} \cdot 12$

Step 8.3.1.1.2

Multiply $\frac{1}{- \frac{\sqrt{3}}{9}}$ by $\frac{- y^{3} \sqrt{3}}{9}$ .

$\frac{- y^{3} \sqrt{3}}{- \frac{\sqrt{3}}{9} \cdot 9} = \frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}} \cdot 12$

Step 8.3.1.1.3

Multiply $9$ by $- 1$ .

$\frac{- y^{3} \sqrt{3}}{- 9 \frac{\sqrt{3}}{9}} = \frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}} \cdot 12$

Step 8.3.1.1.4

Combine $- 9$ and $\frac{\sqrt{3}}{9}$ .

$\frac{- y^{3} \sqrt{3}}{\frac{- 9 \sqrt{3}}{9}} = \frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}} \cdot 12$

Step 8.3.1.1.5

Cancel the common factor of $- 9$ and $9$ .

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Step 8.3.1.1.5.1

Factor $9$ out of $- 9 \sqrt{3}$ .

$\frac{- y^{3} \sqrt{3}}{\frac{9 (- \sqrt{3})}{9}} = \frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}} \cdot 12$

Step 8.3.1.1.5.2

Cancel the common factors.

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Step 8.3.1.1.5.2.1

Factor $9$ out of $9$ .

$\frac{- y^{3} \sqrt{3}}{\frac{9 (- \sqrt{3})}{9 (1)}} = \frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}} \cdot 12$

Step 8.3.1.1.5.2.2

Cancel the common factor.

$\frac{- y^{3} \sqrt{3}}{\frac{9 (- \sqrt{3})}{9 \cdot 1}} = \frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}} \cdot 12$

Step 8.3.1.1.5.2.3

Rewrite the expression.

$\frac{- y^{3} \sqrt{3}}{\frac{- \sqrt{3}}{1}} = \frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}} \cdot 12$

Step 8.3.1.1.5.2.4

Divide $- \sqrt{3}$ by $1$ .

$\frac{- y^{3} \sqrt{3}}{- \sqrt{3}} = \frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}} \cdot 12$

Step 8.3.1.1.6

Dividing two negative values results in a positive value.

$\frac{y^{3} \sqrt{3}}{\sqrt{3}} = \frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}} \cdot 12$

Step 8.3.1.1.7

Cancel the common factor of $\sqrt{3}$ .

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Step 8.3.1.1.7.1

Cancel the common factor.

$\frac{y^{3} \sqrt{3}}{\sqrt{3}} = \frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}} \cdot 12$

Step 8.3.1.1.7.2

Divide $y^{3}$ by $1$ .

$y^{3} = \frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}} \cdot 12$

Step 8.3.2

Simplify the right side.

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Step 8.3.2.1

Simplify $\frac{1}{2 \cdot - 1 \frac{\sqrt{3}}{9}} \cdot 12$ .

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Step 8.3.2.1.1

Simplify the denominator.

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Step 8.3.2.1.1.1

Multiply $2$ by $- 1$ .

$y^{3} = \frac{1}{- 2 \frac{\sqrt{3}}{9}} \cdot 12$

Step 8.3.2.1.1.2

Combine $- 2$ and $\frac{\sqrt{3}}{9}$ .

$y^{3} = \frac{1}{\frac{- 2 \sqrt{3}}{9}} \cdot 12$

Step 8.3.2.1.2

Reduce the expression by cancelling the common factors.

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Step 8.3.2.1.2.1

Move the negative in front of the fraction.

$y^{3} = \frac{1}{- \frac{2 \sqrt{3}}{9}} \cdot 12$

Step 8.3.2.1.2.2

Cancel the common factor of $1$ and $- 1$ .

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Step 8.3.2.1.2.2.1

Rewrite $1$ as $- 1 (- 1)$ .

$y^{3} = \frac{- 1 (- 1)}{- \frac{2 \sqrt{3}}{9}} \cdot 12$

Step 8.3.2.1.2.2.2

Move the negative in front of the fraction.

$y^{3} = - \frac{1}{\frac{2 \sqrt{3}}{9}} \cdot 12$

Step 8.3.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$y^{3} = - (1 \frac{9}{2 \sqrt{3}}) \cdot 12$

Step 8.3.2.1.4

Multiply $\frac{9}{2 \sqrt{3}}$ by $1$ .

$y^{3} = - \frac{9}{2 \sqrt{3}} \cdot 12$

Step 8.3.2.1.5

Cancel the common factor of $2$ .

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Step 8.3.2.1.5.1

Move the leading negative in $- \frac{9}{2 \sqrt{3}}$ into the numerator.

$y^{3} = \frac{- 9}{2 \sqrt{3}} \cdot 12$

Step 8.3.2.1.5.2

Factor $2$ out of $2 \sqrt{3}$ .

$y^{3} = \frac{- 9}{2 (\sqrt{3})} \cdot 12$

Step 8.3.2.1.5.3

Factor $2$ out of $12$ .

$y^{3} = \frac{- 9}{2 (\sqrt{3})} \cdot (2 (6))$

Step 8.3.2.1.5.4

Cancel the common factor.

$y^{3} = \frac{- 9}{2 \sqrt{3}} \cdot (2 \cdot 6)$

Step 8.3.2.1.5.5

Rewrite the expression.

$y^{3} = \frac{- 9}{\sqrt{3}} \cdot 6$

Step 8.3.2.1.6

Combine $\frac{- 9}{\sqrt{3}}$ and $6$ .

$y^{3} = \frac{- 9 \cdot 6}{\sqrt{3}}$

Step 8.3.2.1.7

Simplify the expression.

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Step 8.3.2.1.7.1

Multiply $- 9$ by $6$ .

$y^{3} = \frac{- 54}{\sqrt{3}}$

Step 8.3.2.1.7.2

Move the negative in front of the fraction.

$y^{3} = - \frac{54}{\sqrt{3}}$

Step 8.3.2.1.8

Multiply $\frac{54}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y^{3} = - (\frac{54}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 8.3.2.1.9

Combine and simplify the denominator.

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Step 8.3.2.1.9.1

Multiply $\frac{54}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y^{3} = - \frac{54 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 8.3.2.1.9.2

Raise $\sqrt{3}$ to the power of $1$ .

$y^{3} = - \frac{54 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 8.3.2.1.9.3

Raise $\sqrt{3}$ to the power of $1$ .

$y^{3} = - \frac{54 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 8.3.2.1.9.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y^{3} = - \frac{54 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 8.3.2.1.9.5

Add $1$ and $1$ .

$y^{3} = - \frac{54 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 8.3.2.1.9.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 8.3.2.1.9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$y^{3} = - \frac{54 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 8.3.2.1.9.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y^{3} = - \frac{54 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 8.3.2.1.9.6.3

Combine $\frac{1}{2}$ and $2$ .

$y^{3} = - \frac{54 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 8.3.2.1.9.6.4

Cancel the common factor of $2$ .

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Step 8.3.2.1.9.6.4.1

Cancel the common factor.

$y^{3} = - \frac{54 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 8.3.2.1.9.6.4.2

Rewrite the expression.

$y^{3} = - \frac{54 \sqrt{3}}{3^{1}}$

Step 8.3.2.1.9.6.5

Evaluate the exponent.

$y^{3} = - \frac{54 \sqrt{3}}{3}$

Step 8.3.2.1.10

Cancel the common factor of $54$ and $3$ .

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Step 8.3.2.1.10.1

Factor $3$ out of $54 \sqrt{3}$ .

$y^{3} = - \frac{3 (18 \sqrt{3})}{3}$

Step 8.3.2.1.10.2

Cancel the common factors.

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Step 8.3.2.1.10.2.1

Factor $3$ out of $3$ .

$y^{3} = - \frac{3 (18 \sqrt{3})}{3 (1)}$

Step 8.3.2.1.10.2.2

Cancel the common factor.

$y^{3} = - \frac{3 (18 \sqrt{3})}{3 \cdot 1}$

Step 8.3.2.1.10.2.3

Rewrite the expression.

$y^{3} = - \frac{18 \sqrt{3}}{1}$

Step 8.3.2.1.10.2.4

Divide $18 \sqrt{3}$ by $1$ .

$y^{3} = - (18 \sqrt{3})$

Step 8.3.2.1.11

Multiply $18$ by $- 1$ .

$y^{3} = - 18 \sqrt{3}$

Step 8.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[3]{- 18 \sqrt{3}}$

Step 8.5

Simplify $\sqrt[3]{- 18 \sqrt{3}}$ .

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Step 8.5.1

Rewrite $- 18 \sqrt{3}$ as ${({(- 1)}^{3})}^{3} \cdot (18 \sqrt{3})$ .

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Step 8.5.1.1

Rewrite $- 18$ as $- 1 (18)$ .

$y = \sqrt[3]{- 1 (18) \sqrt{3}}$

Step 8.5.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$y = \sqrt[3]{{(- 1)}^{3} \cdot 18 \sqrt{3}}$

Step 8.5.1.3

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$y = \sqrt[3]{{({(- 1)}^{3})}^{3} \cdot 18 \sqrt{3}}$

Step 8.5.1.4

Add parentheses.

$y = \sqrt[3]{{({(- 1)}^{3})}^{3} \cdot (18 \sqrt{3})}$

Step 8.5.2

Pull terms out from under the radical.

$y = {(- 1)}^{3} \sqrt[3]{18 \sqrt{3}}$

Step 8.5.3

Simplify the expression.

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Step 8.5.3.1

Raise $- 1$ to the power of $3$ .

$y = - 1 \sqrt[3]{18 \sqrt{3}}$

Step 8.5.3.2

Rewrite $- 1 \sqrt[3]{18 \sqrt{3}}$ as $- \sqrt[3]{18 \sqrt{3}}$ .

$y = - \sqrt[3]{18 \sqrt{3}}$

Step 9

Solve for $y$ .

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Step 9.1

Simplify ${(\frac{y (3 - \sqrt{3})}{3})}^{3} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2}$ .

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Step 9.1.1

Simplify each term.

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Step 9.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 9.1.1.1.1

Apply the product rule to $\frac{y (3 - \sqrt{3})}{3}$ .

$\frac{{(y (3 - \sqrt{3}))}^{3}}{3^{3}} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.1.2

Apply the product rule to $y (3 - \sqrt{3})$ .

$\frac{y^{3} {(3 - \sqrt{3})}^{3}}{3^{3}} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.2

Raise $3$ to the power of $3$ .

$\frac{y^{3} {(3 - \sqrt{3})}^{3}}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.3

Use the Binomial Theorem.

$\frac{y^{3} (3^{3} + 3 \cdot 3^{2} (- \sqrt{3}) + 3 \cdot 3 {(- \sqrt{3})}^{2} + {(- \sqrt{3})}^{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4

Simplify each term.

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Step 9.1.1.4.1

Raise $3$ to the power of $3$ .

$\frac{y^{3} (27 + 3 \cdot 3^{2} (- \sqrt{3}) + 3 \cdot 3 {(- \sqrt{3})}^{2} + {(- \sqrt{3})}^{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.2

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 9.1.1.4.2.1

Multiply $3$ by $3^{2}$ .

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Step 9.1.1.4.2.1.1

Raise $3$ to the power of $1$ .

$\frac{y^{3} (27 + 3^{1} \cdot 3^{2} (- \sqrt{3}) + 3 \cdot 3 {(- \sqrt{3})}^{2} + {(- \sqrt{3})}^{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{y^{3} (27 + 3^{1 + 2} (- \sqrt{3}) + 3 \cdot 3 {(- \sqrt{3})}^{2} + {(- \sqrt{3})}^{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.2.2

Add $1$ and $2$ .

$\frac{y^{3} (27 + 3^{3} (- \sqrt{3}) + 3 \cdot 3 {(- \sqrt{3})}^{2} + {(- \sqrt{3})}^{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.3

Raise $3$ to the power of $3$ .

$\frac{y^{3} (27 + 27 (- \sqrt{3}) + 3 \cdot 3 {(- \sqrt{3})}^{2} + {(- \sqrt{3})}^{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.4

Multiply $- 1$ by $27$ .

$\frac{y^{3} (27 - 27 \sqrt{3} + 3 \cdot 3 {(- \sqrt{3})}^{2} + {(- \sqrt{3})}^{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.5

Multiply $3$ by $3$ .

$\frac{y^{3} (27 - 27 \sqrt{3} + 9 {(- \sqrt{3})}^{2} + {(- \sqrt{3})}^{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.6

Apply the product rule to $- \sqrt{3}$ .

$\frac{y^{3} (27 - 27 \sqrt{3} + 9 ({(- 1)}^{2} {\sqrt{3}}^{2}) + {(- \sqrt{3})}^{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.7

Raise $- 1$ to the power of $2$ .

$\frac{y^{3} (27 - 27 \sqrt{3} + 9 (1 {\sqrt{3}}^{2}) + {(- \sqrt{3})}^{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.8

Multiply ${\sqrt{3}}^{2}$ by $1$ .

$\frac{y^{3} (27 - 27 \sqrt{3} + 9 {\sqrt{3}}^{2} + {(- \sqrt{3})}^{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.9

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 9.1.1.4.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{y^{3} (27 - 27 \sqrt{3} + 9 {(3^{\frac{1}{2}})}^{2} + {(- \sqrt{3})}^{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.9.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{y^{3} (27 - 27 \sqrt{3} + 9 \cdot 3^{\frac{1}{2} \cdot 2} + {(- \sqrt{3})}^{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.9.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{y^{3} (27 - 27 \sqrt{3} + 9 \cdot 3^{\frac{2}{2}} + {(- \sqrt{3})}^{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.9.4

Cancel the common factor of $2$ .

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Step 9.1.1.4.9.4.1

Cancel the common factor.

$\frac{y^{3} (27 - 27 \sqrt{3} + 9 \cdot 3^{\frac{2}{2}} + {(- \sqrt{3})}^{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.9.4.2

Rewrite the expression.

$\frac{y^{3} (27 - 27 \sqrt{3} + 9 \cdot 3^{1} + {(- \sqrt{3})}^{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.9.5

Evaluate the exponent.

$\frac{y^{3} (27 - 27 \sqrt{3} + 9 \cdot 3 + {(- \sqrt{3})}^{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.10

Multiply $9$ by $3$ .

$\frac{y^{3} (27 - 27 \sqrt{3} + 27 + {(- \sqrt{3})}^{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.11

Apply the product rule to $- \sqrt{3}$ .

$\frac{y^{3} (27 - 27 \sqrt{3} + 27 + {(- 1)}^{3} {\sqrt{3}}^{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.12

Raise $- 1$ to the power of $3$ .

$\frac{y^{3} (27 - 27 \sqrt{3} + 27 - {\sqrt{3}}^{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.13

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$\frac{y^{3} (27 - 27 \sqrt{3} + 27 - \sqrt{3^{3}})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.14

Raise $3$ to the power of $3$ .

$\frac{y^{3} (27 - 27 \sqrt{3} + 27 - \sqrt{27})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.15

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 9.1.1.4.15.1

Factor $9$ out of $27$ .

$\frac{y^{3} (27 - 27 \sqrt{3} + 27 - \sqrt{9 (3)})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.15.2

Rewrite $9$ as $3^{2}$ .

$\frac{y^{3} (27 - 27 \sqrt{3} + 27 - \sqrt{3^{2} \cdot 3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.16

Pull terms out from under the radical.

$\frac{y^{3} (27 - 27 \sqrt{3} + 27 - (3 \sqrt{3}))}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.4.17

Multiply $3$ by $- 1$ .

$\frac{y^{3} (27 - 27 \sqrt{3} + 27 - 3 \sqrt{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.5

Add $27$ and $27$ .

$\frac{y^{3} (54 - 27 \sqrt{3} - 3 \sqrt{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.6

Subtract $3 \sqrt{3}$ from $- 27 \sqrt{3}$ .

$\frac{y^{3} (54 - 30 \sqrt{3})}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.7

Cancel the common factor of $54 - 30 \sqrt{3}$ and $27$ .

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Step 9.1.1.7.1

Factor $3$ out of $y^{3} (54 - 30 \sqrt{3})$ .

$\frac{3 (y^{3} (18 - 10 \sqrt{3}))}{27} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.7.2

Cancel the common factors.

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Step 9.1.1.7.2.1

Factor $3$ out of $27$ .

$\frac{3 (y^{3} (18 - 10 \sqrt{3}))}{3 (9)} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.7.2.2

Cancel the common factor.

$\frac{3 (y^{3} (18 - 10 \sqrt{3}))}{3 \cdot 9} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.7.2.3

Rewrite the expression.

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 3 {(\frac{y (3 - \sqrt{3})}{3})}^{2} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.8

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 9.1.1.8.1

Apply the product rule to $\frac{y (3 - \sqrt{3})}{3}$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 3 \frac{{(y (3 - \sqrt{3}))}^{2}}{3^{2}} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.8.2

Apply the product rule to $y (3 - \sqrt{3})$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 3 \frac{y^{2} {(3 - \sqrt{3})}^{2}}{3^{2}} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.9

Raise $3$ to the power of $2$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 3 \frac{y^{2} {(3 - \sqrt{3})}^{2}}{9} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.10

Cancel the common factor of $3$ .

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Step 9.1.1.10.1

Factor $3$ out of $- 3$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} + 3 (- 1) \frac{y^{2} {(3 - \sqrt{3})}^{2}}{9} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.10.2

Factor $3$ out of $9$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} + 3 \cdot - 1 \frac{y^{2} {(3 - \sqrt{3})}^{2}}{3 \cdot 3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.10.3

Cancel the common factor.

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} + 3 \cdot - 1 \frac{y^{2} {(3 - \sqrt{3})}^{2}}{3 \cdot 3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.10.4

Rewrite the expression.

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} {(3 - \sqrt{3})}^{2}}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.11

Rewrite ${(3 - \sqrt{3})}^{2}$ as $(3 - \sqrt{3}) (3 - \sqrt{3})$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} ((3 - \sqrt{3}) (3 - \sqrt{3}))}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.12

Expand $(3 - \sqrt{3}) (3 - \sqrt{3})$ using the FOIL Method.

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Step 9.1.1.12.1

Apply the distributive property.

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} (3 (3 - \sqrt{3}) - \sqrt{3} (3 - \sqrt{3}))}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.12.2

Apply the distributive property.

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} (3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} (3 - \sqrt{3}))}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.12.3

Apply the distributive property.

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} (3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3}))}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.13

Simplify and combine like terms.

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Step 9.1.1.13.1

Simplify each term.

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Step 9.1.1.13.1.1

Multiply $3$ by $3$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} (9 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3}))}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.13.1.2

Multiply $- 1$ by $3$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} (9 - 3 \sqrt{3} - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3}))}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.13.1.3

Multiply $3$ by $- 1$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} (9 - 3 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3}))}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.13.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

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Step 9.1.1.13.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} (9 - 3 \sqrt{3} - 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3})}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.13.1.4.2

Multiply $\sqrt{3}$ by $1$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} (9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3})}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.13.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} (9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3})}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.13.1.4.4

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} (9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1})}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.13.1.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} (9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1 + 1})}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.13.1.4.6

Add $1$ and $1$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} (9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{2})}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.13.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 9.1.1.13.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} (9 - 3 \sqrt{3} - 3 \sqrt{3} + {(3^{\frac{1}{2}})}^{2})}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.13.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} (9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{1}{2} \cdot 2})}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.13.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} (9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}})}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.13.1.5.4

Cancel the common factor of $2$ .

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Step 9.1.1.13.1.5.4.1

Cancel the common factor.

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} (9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}})}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.13.1.5.4.2

Rewrite the expression.

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} (9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{1})}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.13.1.5.5

Evaluate the exponent.

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} (9 - 3 \sqrt{3} - 3 \sqrt{3} + 3)}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.13.2

Add $9$ and $3$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} (12 - 3 \sqrt{3} - 3 \sqrt{3})}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.13.3

Subtract $3 \sqrt{3}$ from $- 3 \sqrt{3}$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} (12 - 6 \sqrt{3})}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.14

Cancel the common factor of $12 - 6 \sqrt{3}$ and $3$ .

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Step 9.1.1.14.1

Factor $3$ out of $y^{2} (12 - 6 \sqrt{3})$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{3 (y^{2} (4 - 2 \sqrt{3}))}{3} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.14.2

Cancel the common factors.

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Step 9.1.1.14.2.1

Factor $3$ out of $3$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{3 (y^{2} (4 - 2 \sqrt{3}))}{3 (1)} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.14.2.2

Cancel the common factor.

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{3 (y^{2} (4 - 2 \sqrt{3}))}{3 \cdot 1} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.14.2.3

Rewrite the expression.

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 \frac{y^{2} (4 - 2 \sqrt{3})}{1} y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.14.2.4

Divide $y^{2} (4 - 2 \sqrt{3})$ by $1$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 (y^{2} (4 - 2 \sqrt{3})) y + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.15

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 9.1.1.15.1

Move $y$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 (y \cdot y^{2} (4 - 2 \sqrt{3})) + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.15.2

Multiply $y$ by $y^{2}$ .

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Step 9.1.1.15.2.1

Raise $y$ to the power of $1$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 (y^{1} y^{2} (4 - 2 \sqrt{3})) + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.15.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 (y^{1 + 2} (4 - 2 \sqrt{3})) + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.15.3

Add $1$ and $2$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 (y^{3} (4 - 2 \sqrt{3})) + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.16

Apply the distributive property.

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 (y^{3} \cdot 4 + y^{3} (- 2 \sqrt{3})) + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.17

Move $4$ to the left of $y^{3}$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 (4 \cdot y^{3} + y^{3} (- 2 \sqrt{3})) + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.18

Apply the distributive property.

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 1 (4 y^{3}) - 1 (y^{3} (- 2 \sqrt{3})) + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.19

Multiply $4$ by $- 1$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 4 y^{3} - 1 (y^{3} (- 2 \sqrt{3})) + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.20

Multiply $- 2$ by $- 1$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 4 y^{3} + 2 (y^{3} (\sqrt{3})) + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.21

Remove parentheses.

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 4 y^{3} + 2 y^{3} \sqrt{3} + 2 (\frac{y (3 - \sqrt{3})}{3}) \cdot y^{2} = 12$

Step 9.1.1.22

Combine $2$ and $\frac{y (3 - \sqrt{3})}{3}$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 4 y^{3} + 2 y^{3} \sqrt{3} + \frac{2 (y (3 - \sqrt{3}))}{3} \cdot y^{2} = 12$

Step 9.1.1.23

Multiply $\frac{2 y (3 - \sqrt{3})}{3} y^{2}$ .

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Step 9.1.1.23.1

Combine $\frac{2 y (3 - \sqrt{3})}{3}$ and $y^{2}$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 4 y^{3} + 2 y^{3} \sqrt{3} + \frac{2 y (3 - \sqrt{3}) y^{2}}{3} = 12$

Step 9.1.1.23.2

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 9.1.1.23.2.1

Move $y^{2}$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 4 y^{3} + 2 y^{3} \sqrt{3} + \frac{2 (y^{2} y) (3 - \sqrt{3})}{3} = 12$

Step 9.1.1.23.2.2

Multiply $y^{2}$ by $y$ .

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Step 9.1.1.23.2.2.1

Raise $y$ to the power of $1$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 4 y^{3} + 2 y^{3} \sqrt{3} + \frac{2 (y^{2} y^{1}) (3 - \sqrt{3})}{3} = 12$

Step 9.1.1.23.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 4 y^{3} + 2 y^{3} \sqrt{3} + \frac{2 y^{2 + 1} (3 - \sqrt{3})}{3} = 12$

Step 9.1.1.23.2.3

Add $2$ and $1$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 4 y^{3} + 2 y^{3} \sqrt{3} + \frac{2 y^{3} (3 - \sqrt{3})}{3} = 12$

Step 9.1.2

To write $- 4 y^{3}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} - 4 y^{3} \cdot \frac{9}{9} + 2 y^{3} \sqrt{3} + \frac{2 y^{3} (3 - \sqrt{3})}{3} = 12$

Step 9.1.3

Simplify terms.

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Step 9.1.3.1

Combine $- 4 y^{3}$ and $\frac{9}{9}$ .

$\frac{y^{3} (18 - 10 \sqrt{3})}{9} + \frac{- 4 y^{3} \cdot 9}{9} + 2 y^{3} \sqrt{3} + \frac{2 y^{3} (3 - \sqrt{3})}{3} = 12$

Step 9.1.3.2

Combine the numerators over the common denominator.

$\frac{y^{3} (18 - 10 \sqrt{3}) - 4 y^{3} \cdot 9}{9} + 2 y^{3} \sqrt{3} + \frac{2 y^{3} (3 - \sqrt{3})}{3} = 12$

Step 9.1.4

Simplify the numerator.

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Step 9.1.4.1

Factor $y^{3}$ out of $y^{3} (18 - 10 \sqrt{3}) - 4 y^{3} \cdot 9$ .

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Step 9.1.4.1.1

Factor $y^{3}$ out of $- 4 y^{3} \cdot 9$ .

$\frac{y^{3} (18 - 10 \sqrt{3}) + y^{3} (- 4 \cdot 9)}{9} + 2 y^{3} \sqrt{3} + \frac{2 y^{3} (3 - \sqrt{3})}{3} = 12$

Step 9.1.4.1.2

Factor $y^{3}$ out of $y^{3} (18 - 10 \sqrt{3}) + y^{3} (- 4 \cdot 9)$ .

$\frac{y^{3} (18 - 10 \sqrt{3} - 4 \cdot 9)}{9} + 2 y^{3} \sqrt{3} + \frac{2 y^{3} (3 - \sqrt{3})}{3} = 12$

Step 9.1.4.2

Multiply $- 4$ by $9$ .

$\frac{y^{3} (18 - 10 \sqrt{3} - 36)}{9} + 2 y^{3} \sqrt{3} + \frac{2 y^{3} (3 - \sqrt{3})}{3} = 12$

Step 9.1.4.3

Subtract $36$ from $18$ .

$\frac{y^{3} (- 18 - 10 \sqrt{3})}{9} + 2 y^{3} \sqrt{3} + \frac{2 y^{3} (3 - \sqrt{3})}{3} = 12$

Step 9.1.5

Find the common denominator.

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Step 9.1.5.1

Write $2 y^{3} \sqrt{3}$ as a fraction with denominator $1$ .

$\frac{y^{3} (- 18 - 10 \sqrt{3})}{9} + \frac{2 y^{3} \sqrt{3}}{1} + \frac{2 y^{3} (3 - \sqrt{3})}{3} = 12$

Step 9.1.5.2

Multiply $\frac{2 y^{3} \sqrt{3}}{1}$ by $\frac{9}{9}$ .

$\frac{y^{3} (- 18 - 10 \sqrt{3})}{9} + \frac{2 y^{3} \sqrt{3}}{1} \cdot \frac{9}{9} + \frac{2 y^{3} (3 - \sqrt{3})}{3} = 12$

Step 9.1.5.3

Multiply $\frac{2 y^{3} \sqrt{3}}{1}$ by $\frac{9}{9}$ .

$\frac{y^{3} (- 18 - 10 \sqrt{3})}{9} + \frac{2 y^{3} \sqrt{3} \cdot 9}{9} + \frac{2 y^{3} (3 - \sqrt{3})}{3} = 12$

Step 9.1.5.4

Multiply $\frac{2 y^{3} (3 - \sqrt{3})}{3}$ by $\frac{3}{3}$ .

$\frac{y^{3} (- 18 - 10 \sqrt{3})}{9} + \frac{2 y^{3} \sqrt{3} \cdot 9}{9} + \frac{2 y^{3} (3 - \sqrt{3})}{3} \cdot \frac{3}{3} = 12$

Step 9.1.5.5

Multiply $\frac{2 y^{3} (3 - \sqrt{3})}{3}$ by $\frac{3}{3}$ .

$\frac{y^{3} (- 18 - 10 \sqrt{3})}{9} + \frac{2 y^{3} \sqrt{3} \cdot 9}{9} + \frac{2 y^{3} (3 - \sqrt{3}) \cdot 3}{3 \cdot 3} = 12$

Step 9.1.5.6

Multiply $3$ by $3$ .

$\frac{y^{3} (- 18 - 10 \sqrt{3})}{9} + \frac{2 y^{3} \sqrt{3} \cdot 9}{9} + \frac{2 y^{3} (3 - \sqrt{3}) \cdot 3}{9} = 12$

Step 9.1.6

Combine the numerators over the common denominator.

$\frac{y^{3} (- 18 - 10 \sqrt{3}) + 2 y^{3} \sqrt{3} \cdot 9 + 2 y^{3} (3 - \sqrt{3}) \cdot 3}{9} = 12$

Step 9.1.7

Simplify each term.

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Step 9.1.7.1

Apply the distributive property.

$\frac{y^{3} \cdot - 18 + y^{3} (- 10 \sqrt{3}) + 2 y^{3} \sqrt{3} \cdot 9 + 2 y^{3} (3 - \sqrt{3}) \cdot 3}{9} = 12$

Step 9.1.7.2

Move $- 18$ to the left of $y^{3}$ .

$\frac{- 18 \cdot y^{3} + y^{3} (- 10 \sqrt{3}) + 2 y^{3} \sqrt{3} \cdot 9 + 2 y^{3} (3 - \sqrt{3}) \cdot 3}{9} = 12$

Step 9.1.7.3

Multiply $9$ by $2$ .

$\frac{- 18 y^{3} + y^{3} (- 10 \sqrt{3}) + 18 y^{3} \sqrt{3} + 2 y^{3} (3 - \sqrt{3}) \cdot 3}{9} = 12$

Step 9.1.7.4

Apply the distributive property.

$\frac{- 18 y^{3} + y^{3} (- 10 \sqrt{3}) + 18 y^{3} \sqrt{3} + (2 y^{3} \cdot 3 + 2 y^{3} (- \sqrt{3})) \cdot 3}{9} = 12$

Step 9.1.7.5

Multiply $3$ by $2$ .

$\frac{- 18 y^{3} + y^{3} (- 10 \sqrt{3}) + 18 y^{3} \sqrt{3} + (6 y^{3} + 2 y^{3} (- \sqrt{3})) \cdot 3}{9} = 12$

Step 9.1.7.6

Multiply $- 1$ by $2$ .

$\frac{- 18 y^{3} + y^{3} (- 10 \sqrt{3}) + 18 y^{3} \sqrt{3} + (6 y^{3} - 2 y^{3} \sqrt{3}) \cdot 3}{9} = 12$

Step 9.1.7.7

Apply the distributive property.

$\frac{- 18 y^{3} + y^{3} (- 10 \sqrt{3}) + 18 y^{3} \sqrt{3} + 6 y^{3} \cdot 3 - 2 y^{3} \sqrt{3} \cdot 3}{9} = 12$

Step 9.1.7.8

Multiply $3$ by $6$ .

$\frac{- 18 y^{3} + y^{3} (- 10 \sqrt{3}) + 18 y^{3} \sqrt{3} + 18 y^{3} - 2 y^{3} \sqrt{3} \cdot 3}{9} = 12$

Step 9.1.7.9

Multiply $3$ by $- 2$ .

$\frac{- 18 y^{3} + y^{3} (- 10 \sqrt{3}) + 18 y^{3} \sqrt{3} + 18 y^{3} - 6 y^{3} \sqrt{3}}{9} = 12$

Step 9.1.8

Combine the opposite terms in $- 18 y^{3} + y^{3} (- 10 \sqrt{3}) + 18 y^{3} \sqrt{3} + 18 y^{3} - 6 y^{3} \sqrt{3}$ .

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Step 9.1.8.1

Add $- 18 y^{3}$ and $18 y^{3}$ .

$\frac{y^{3} (- 10 \sqrt{3}) + 18 y^{3} \sqrt{3} + 0 - 6 y^{3} \sqrt{3}}{9} = 12$

Step 9.1.8.2

Add $y^{3} (- 10 \sqrt{3}) + 18 y^{3} \sqrt{3}$ and $0$ .

$\frac{y^{3} (- 10 \sqrt{3}) + 18 y^{3} \sqrt{3} - 6 y^{3} \sqrt{3}}{9} = 12$

Step 9.1.9

Add $y^{3} (- 10 \sqrt{3})$ and $18 y^{3} \sqrt{3}$ .

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Step 9.1.9.1

Reorder $y^{3}$ and $- 10$ .

$\frac{- 10 \cdot y^{3} \sqrt{3} + 18 y^{3} \sqrt{3} - 6 y^{3} \sqrt{3}}{9} = 12$

Step 9.1.9.2

Add $- 10 \cdot y^{3} \sqrt{3}$ and $18 y^{3} \sqrt{3}$ .

$\frac{8 y^{3} \sqrt{3} - 6 y^{3} \sqrt{3}}{9} = 12$

Step 9.1.10

Subtract $6 y^{3} \sqrt{3}$ from $8 y^{3} \sqrt{3}$ .

$\frac{2 y^{3} \sqrt{3}}{9} = 12$

Step 9.2

Multiply both sides of the equation by $\frac{1}{2 \frac{\sqrt{3}}{9}}$ .

$\frac{1}{2 \frac{\sqrt{3}}{9}} \cdot \frac{2 y^{3} \sqrt{3}}{9} = \frac{1}{2 \frac{\sqrt{3}}{9}} \cdot 12$

Step 9.3

Simplify both sides of the equation.

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Step 9.3.1

Simplify the left side.

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Step 9.3.1.1

Simplify $\frac{1}{2 \frac{\sqrt{3}}{9}} \cdot \frac{2 y^{3} \sqrt{3}}{9}$ .

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Step 9.3.1.1.1

Combine.

$\frac{1 (2 y^{3} \sqrt{3})}{2 \frac{\sqrt{3}}{9} \cdot 9} = \frac{1}{2 \frac{\sqrt{3}}{9}} \cdot 12$

Step 9.3.1.1.2

Cancel the common factor of $2$ .

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Step 9.3.1.1.2.1

Cancel the common factor.

$\frac{1 (2 y^{3} \sqrt{3})}{2 \frac{\sqrt{3}}{9} \cdot 9} = \frac{1}{2 \frac{\sqrt{3}}{9}} \cdot 12$

Step 9.3.1.1.2.2

Rewrite the expression.

$\frac{1 (y^{3} \sqrt{3})}{\frac{\sqrt{3}}{9} \cdot 9} = \frac{1}{2 \frac{\sqrt{3}}{9}} \cdot 12$

Step 9.3.1.1.3

Multiply $y^{3}$ by $1$ .

$\frac{y^{3} \sqrt{3}}{\frac{\sqrt{3}}{9} \cdot 9} = \frac{1}{2 \frac{\sqrt{3}}{9}} \cdot 12$

Step 9.3.1.1.4

Combine $\frac{\sqrt{3}}{9}$ and $9$ .

$\frac{y^{3} \sqrt{3}}{\frac{\sqrt{3} \cdot 9}{9}} = \frac{1}{2 \frac{\sqrt{3}}{9}} \cdot 12$

Step 9.3.1.1.5

Move $9$ to the left of $\sqrt{3}$ .

$\frac{y^{3} \sqrt{3}}{\frac{9 \sqrt{3}}{9}} = \frac{1}{2 \frac{\sqrt{3}}{9}} \cdot 12$

Step 9.3.1.1.6

Reduce the expression by cancelling the common factors.

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Step 9.3.1.1.6.1

Reduce the expression $\frac{9 \sqrt{3}}{9}$ by cancelling the common factors.

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Step 9.3.1.1.6.1.1

Cancel the common factor.

$\frac{y^{3} \sqrt{3}}{\frac{9 \sqrt{3}}{9}} = \frac{1}{2 \frac{\sqrt{3}}{9}} \cdot 12$

Step 9.3.1.1.6.1.2

Rewrite the expression.

$\frac{y^{3} \sqrt{3}}{\frac{\sqrt{3}}{1}} = \frac{1}{2 \frac{\sqrt{3}}{9}} \cdot 12$

Step 9.3.1.1.6.2

Divide $\sqrt{3}$ by $1$ .

$\frac{y^{3} \sqrt{3}}{\sqrt{3}} = \frac{1}{2 \frac{\sqrt{3}}{9}} \cdot 12$

Step 9.3.1.1.7

Cancel the common factor of $\sqrt{3}$ .

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Step 9.3.1.1.7.1

Cancel the common factor.

$\frac{y^{3} \sqrt{3}}{\sqrt{3}} = \frac{1}{2 \frac{\sqrt{3}}{9}} \cdot 12$

Step 9.3.1.1.7.2

Divide $y^{3}$ by $1$ .

$y^{3} = \frac{1}{2 \frac{\sqrt{3}}{9}} \cdot 12$

Step 9.3.2

Simplify the right side.

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Step 9.3.2.1

Simplify $\frac{1}{2 \frac{\sqrt{3}}{9}} \cdot 12$ .

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Step 9.3.2.1.1

Combine $2$ and $\frac{\sqrt{3}}{9}$ .

$y^{3} = \frac{1}{\frac{2 \sqrt{3}}{9}} \cdot 12$

Step 9.3.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$y^{3} = 1 \frac{9}{2 \sqrt{3}} \cdot 12$

Step 9.3.2.1.3

Multiply $\frac{9}{2 \sqrt{3}}$ by $1$ .

$y^{3} = \frac{9}{2 \sqrt{3}} \cdot 12$

Step 9.3.2.1.4

Cancel the common factor of $2$ .

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Step 9.3.2.1.4.1

Factor $2$ out of $2 \sqrt{3}$ .

$y^{3} = \frac{9}{2 (\sqrt{3})} \cdot 12$

Step 9.3.2.1.4.2

Factor $2$ out of $12$ .

$y^{3} = \frac{9}{2 (\sqrt{3})} \cdot (2 (6))$

Step 9.3.2.1.4.3

Cancel the common factor.

$y^{3} = \frac{9}{2 \sqrt{3}} \cdot (2 \cdot 6)$

Step 9.3.2.1.4.4

Rewrite the expression.

$y^{3} = \frac{9}{\sqrt{3}} \cdot 6$

Step 9.3.2.1.5

Combine $\frac{9}{\sqrt{3}}$ and $6$ .

$y^{3} = \frac{9 \cdot 6}{\sqrt{3}}$

Step 9.3.2.1.6

Multiply $9$ by $6$ .

$y^{3} = \frac{54}{\sqrt{3}}$

Step 9.3.2.1.7

Multiply $\frac{54}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y^{3} = \frac{54}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 9.3.2.1.8

Combine and simplify the denominator.

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Step 9.3.2.1.8.1

Multiply $\frac{54}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y^{3} = \frac{54 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 9.3.2.1.8.2

Raise $\sqrt{3}$ to the power of $1$ .

$y^{3} = \frac{54 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 9.3.2.1.8.3

Raise $\sqrt{3}$ to the power of $1$ .

$y^{3} = \frac{54 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 9.3.2.1.8.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y^{3} = \frac{54 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 9.3.2.1.8.5

Add $1$ and $1$ .

$y^{3} = \frac{54 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 9.3.2.1.8.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 9.3.2.1.8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$y^{3} = \frac{54 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 9.3.2.1.8.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y^{3} = \frac{54 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 9.3.2.1.8.6.3

Combine $\frac{1}{2}$ and $2$ .

$y^{3} = \frac{54 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 9.3.2.1.8.6.4

Cancel the common factor of $2$ .

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Step 9.3.2.1.8.6.4.1

Cancel the common factor.

$y^{3} = \frac{54 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 9.3.2.1.8.6.4.2

Rewrite the expression.

$y^{3} = \frac{54 \sqrt{3}}{3^{1}}$

Step 9.3.2.1.8.6.5

Evaluate the exponent.

$y^{3} = \frac{54 \sqrt{3}}{3}$

Step 9.3.2.1.9

Cancel the common factor of $54$ and $3$ .

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Step 9.3.2.1.9.1

Factor $3$ out of $54 \sqrt{3}$ .

$y^{3} = \frac{3 (18 \sqrt{3})}{3}$

Step 9.3.2.1.9.2

Cancel the common factors.

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Step 9.3.2.1.9.2.1

Factor $3$ out of $3$ .

$y^{3} = \frac{3 (18 \sqrt{3})}{3 (1)}$

Step 9.3.2.1.9.2.2

Cancel the common factor.

$y^{3} = \frac{3 (18 \sqrt{3})}{3 \cdot 1}$

Step 9.3.2.1.9.2.3

Rewrite the expression.

$y^{3} = \frac{18 \sqrt{3}}{1}$

Step 9.3.2.1.9.2.4

Divide $18 \sqrt{3}$ by $1$ .

$y^{3} = 18 \sqrt{3}$

Step 9.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[3]{18 \sqrt{3}}$

Step 10

Find the points where $\frac{d y}{d x} = 0$ .

$(\frac{y (3 + \sqrt{3})}{3}, - \sqrt[3]{18 \sqrt{3}})$

$(\frac{y (3 - \sqrt{3})}{3}, \sqrt[3]{18 \sqrt{3}})$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$(\frac{y (3 + \sqrt{3})}{3}, - \sqrt[3]{18 \sqrt{3}}) (\frac{y (3 - \sqrt{3})}{3}, \sqrt[3]{18 \sqrt{3}})$

Interval Notation:

$((\frac{y (3 + \sqrt{3})}{3}, - \sqrt[3]{18 \sqrt{3}}), n >)$

Step 12