Calculus Examples

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

Step 1

Find the first derivative and evaluate at $x = - 3 \sqrt{3}$ and $y = 1$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

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

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

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

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

Step 1.2.2

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

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Step 1.2.2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

$\frac{2}{3} x^{\frac{2}{3} - 1} + \frac{d}{d x} [y^{\frac{2}{3}}]$

Step 1.2.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}} + \frac{d}{d x} [y^{\frac{2}{3}}]$

Step 1.2.2.3

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} + \frac{d}{d x} [y^{\frac{2}{3}}]$

Step 1.2.2.4

Combine the numerators over the common denominator.

$\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}} + \frac{d}{d x} [y^{\frac{2}{3}}]$

Step 1.2.2.5

Simplify the numerator.

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Step 1.2.2.5.1

Multiply $- 1$ by $3$ .

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

Step 1.2.2.5.2

Subtract $3$ from $2$ .

$\frac{2}{3} x^{\frac{- 1}{3}} + \frac{d}{d x} [y^{\frac{2}{3}}]$

Step 1.2.2.6

Move the negative in front of the fraction.

$\frac{2}{3} x^{- \frac{1}{3}} + \frac{d}{d x} [y^{\frac{2}{3}}]$

Step 1.2.3

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

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Step 1.2.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{2}{3}}$ and $g (x) = y$ .

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Step 1.2.3.1.1

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

$\frac{2}{3} x^{- \frac{1}{3}} + \frac{d}{d u} [u^{\frac{2}{3}}] \frac{d}{d x} [y]$

Step 1.2.3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{2}{3}$ .

$\frac{2}{3} x^{- \frac{1}{3}} + \frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} [y]$

Step 1.2.3.1.3

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

$\frac{2}{3} x^{- \frac{1}{3}} + \frac{2}{3} y^{\frac{2}{3} - 1} \frac{d}{d x} [y]$

Step 1.2.3.2

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

$\frac{2}{3} x^{- \frac{1}{3}} + \frac{2}{3} y^{\frac{2}{3} - 1} y'$

Step 1.2.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 1.2.3.4

Combine $- 1$ and $\frac{3}{3}$ .

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

Step 1.2.3.5

Combine the numerators over the common denominator.

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

Step 1.2.3.6

Simplify the numerator.

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Step 1.2.3.6.1

Multiply $- 1$ by $3$ .

$\frac{2}{3} x^{- \frac{1}{3}} + \frac{2}{3} y^{\frac{2 - 3}{3}} y'$

Step 1.2.3.6.2

Subtract $3$ from $2$ .

$\frac{2}{3} x^{- \frac{1}{3}} + \frac{2}{3} y^{\frac{- 1}{3}} y'$

Step 1.2.3.7

Move the negative in front of the fraction.

$\frac{2}{3} x^{- \frac{1}{3}} + \frac{2}{3} y^{- \frac{1}{3}} y'$

Step 1.2.3.8

Combine $\frac{2}{3}$ and $y^{- \frac{1}{3}}$ .

$\frac{2}{3} x^{- \frac{1}{3}} + \frac{2 y^{- \frac{1}{3}}}{3} y'$

Step 1.2.3.9

Combine $\frac{2 y^{- \frac{1}{3}}}{3}$ and $y'$ .

$\frac{2}{3} x^{- \frac{1}{3}} + \frac{2 y^{- \frac{1}{3}} y'}{3}$

Step 1.2.3.10

Move $y^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2}{3} x^{- \frac{1}{3}} + \frac{2 y'}{3 y^{\frac{1}{3}}}$

Step 1.2.4

Simplify.

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Step 1.2.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}} + \frac{2 y'}{3 y^{\frac{1}{3}}}$

Step 1.2.4.2

Multiply $\frac{2}{3}$ by $\frac{1}{x^{\frac{1}{3}}}$ .

$\frac{2}{3 x^{\frac{1}{3}}} + \frac{2 y'}{3 y^{\frac{1}{3}}}$

Step 1.3

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0$

Step 1.4

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

$\frac{2}{3 x^{\frac{1}{3}}} + \frac{2 y'}{3 y^{\frac{1}{3}}} = 0$

Step 1.5

Solve for $y'$ .

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Step 1.5.1

Subtract $\frac{2}{3 x^{\frac{1}{3}}}$ from both sides of the equation.

$\frac{2 y'}{3 y^{\frac{1}{3}}} = - \frac{2}{3 x^{\frac{1}{3}}}$

Step 1.5.2

Multiply both sides by $3 y^{\frac{1}{3}}$ .

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

Step 1.5.3

Simplify.

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Step 1.5.3.1

Simplify the left side.

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Step 1.5.3.1.1

Simplify $\frac{2 y'}{3 y^{\frac{1}{3}}} (3 y^{\frac{1}{3}})$ .

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Step 1.5.3.1.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.5.3.1.1.2

Cancel the common factor of $3$ .

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Step 1.5.3.1.1.2.1

Cancel the common factor.

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

Step 1.5.3.1.1.2.2

Rewrite the expression.

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

Step 1.5.3.1.1.3

Cancel the common factor of $y^{\frac{1}{3}}$ .

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Step 1.5.3.1.1.3.1

Cancel the common factor.

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

Step 1.5.3.1.1.3.2

Rewrite the expression.

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

Step 1.5.3.2

Simplify the right side.

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Step 1.5.3.2.1

Simplify $- \frac{2}{3 x^{\frac{1}{3}}} (3 y^{\frac{1}{3}})$ .

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Step 1.5.3.2.1.1

Cancel the common factor of $3$ .

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Step 1.5.3.2.1.1.1

Move the leading negative in $- \frac{2}{3 x^{\frac{1}{3}}}$ into the numerator.

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

Step 1.5.3.2.1.1.2

Factor $3$ out of $3 x^{\frac{1}{3}}$ .

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

Step 1.5.3.2.1.1.3

Factor $3$ out of $3 y^{\frac{1}{3}}$ .

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

Step 1.5.3.2.1.1.4

Cancel the common factor.

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

Step 1.5.3.2.1.1.5

Rewrite the expression.

$2 y' = \frac{- 2}{x^{\frac{1}{3}}} y^{\frac{1}{3}}$

Step 1.5.3.2.1.2

Combine $\frac{- 2}{x^{\frac{1}{3}}}$ and $y^{\frac{1}{3}}$ .

$2 y' = \frac{- 2 y^{\frac{1}{3}}}{x^{\frac{1}{3}}}$

Step 1.5.3.2.1.3

Move the negative in front of the fraction.

$2 y' = - \frac{2 y^{\frac{1}{3}}}{x^{\frac{1}{3}}}$

Step 1.5.4

Divide each term in $2 y' = - \frac{2 y^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ by $2$ and simplify.

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Step 1.5.4.1

Divide each term in $2 y' = - \frac{2 y^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ by $2$ .

$\frac{2 y'}{2} = \frac{- \frac{2 y^{\frac{1}{3}}}{x^{\frac{1}{3}}}}{2}$

Step 1.5.4.2

Simplify the left side.

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Step 1.5.4.2.1

Cancel the common factor of $2$ .

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Step 1.5.4.2.1.1

Cancel the common factor.

$\frac{2 y'}{2} = \frac{- \frac{2 y^{\frac{1}{3}}}{x^{\frac{1}{3}}}}{2}$

Step 1.5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- \frac{2 y^{\frac{1}{3}}}{x^{\frac{1}{3}}}}{2}$

Step 1.5.4.3

Simplify the right side.

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Step 1.5.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$y' = - \frac{2 y^{\frac{1}{3}}}{x^{\frac{1}{3}}} \cdot \frac{1}{2}$

Step 1.5.4.3.2

Cancel the common factor of $2$ .

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Step 1.5.4.3.2.1

Move the leading negative in $- \frac{2 y^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ into the numerator.

$y' = \frac{- 2 y^{\frac{1}{3}}}{x^{\frac{1}{3}}} \cdot \frac{1}{2}$

Step 1.5.4.3.2.2

Factor $2$ out of $- 2 y^{\frac{1}{3}}$ .

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

Step 1.5.4.3.2.3

Cancel the common factor.

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

Step 1.5.4.3.2.4

Rewrite the expression.

$y' = \frac{- y^{\frac{1}{3}}}{x^{\frac{1}{3}}}$

Step 1.5.4.3.3

Move the negative in front of the fraction.

$y' = - \frac{y^{\frac{1}{3}}}{x^{\frac{1}{3}}}$

Step 1.6

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

$\frac{d y}{d x} = - \frac{y^{\frac{1}{3}}}{x^{\frac{1}{3}}}$

Step 1.7

Evaluate at $x = - 3 \sqrt{3}$ and $y = 1$ .

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Step 1.7.1

Replace the variable $x$ with $- 3 \sqrt{3}$ in the expression.

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

Step 1.7.2

Replace the variable $y$ with $1$ in the expression.

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

Step 1.7.3

One to any power is one.

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

Step 1.7.4

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

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

Step 2

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 2.1

Use the slope $- \frac{1}{{(- 3)}^{\frac{1}{3}} {\sqrt{3}}^{\frac{1}{3}}}$ and a given point $(- 3 \sqrt{3}, 1)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 2.2

Simplify the equation and keep it in point-slope form.

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- \frac{1}{{(- 3)}^{\frac{1}{3}} {\sqrt{3}}^{\frac{1}{3}}} \cdot (x + 3 \sqrt{3})$ .

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Step 2.3.1.1

Rewrite.

$y - 1 = 0 + 0 - \frac{1}{{(- 3)}^{\frac{1}{3}} {\sqrt{3}}^{\frac{1}{3}}} \cdot (x + 3 \sqrt{3})$

Step 2.3.1.2

Simplify by adding zeros.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.1.4

Combine $x$ and $\frac{1}{{(- 3)}^{\frac{1}{3}} {\sqrt{3}}^{\frac{1}{3}}}$ .

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

Step 2.3.1.5

Multiply $- \frac{1}{{(- 3)}^{\frac{1}{3}} {\sqrt{3}}^{\frac{1}{3}}} (3 \sqrt{3})$ .

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Step 2.3.1.5.1

Multiply $3$ by $- 1$ .

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

Step 2.3.1.5.2

Combine $- 3$ and $\frac{1}{{(- 3)}^{\frac{1}{3}} {\sqrt{3}}^{\frac{1}{3}}}$ .

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

Step 2.3.1.5.3

Combine $\frac{- 3}{{(- 3)}^{\frac{1}{3}} {\sqrt{3}}^{\frac{1}{3}}}$ and $\sqrt{3}$ .

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

Step 2.3.1.6

Simplify each term.

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Step 2.3.1.6.1

Move ${(- 3)}^{\frac{1}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 2.3.1.6.2

Move ${\sqrt{3}}^{\frac{1}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 2.3.1.6.3

Multiply $- 3$ by ${(- 3)}^{- \frac{1}{3}}$ by adding the exponents.

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Step 2.3.1.6.3.1

Move ${(- 3)}^{- \frac{1}{3}}$ .

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

Step 2.3.1.6.3.2

Multiply ${(- 3)}^{- \frac{1}{3}}$ by $- 3$ .

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Step 2.3.1.6.3.2.1

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

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

Step 2.3.1.6.3.2.2

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

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

Step 2.3.1.6.3.3

Write $1$ as a fraction with a common denominator.

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

Step 2.3.1.6.3.4

Combine the numerators over the common denominator.

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

Step 2.3.1.6.3.5

Add $- 1$ and $3$ .

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

Step 2.3.1.6.4

Multiply $\sqrt{3}$ by ${\sqrt{3}}^{- \frac{1}{3}}$ by adding the exponents.

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Step 2.3.1.6.4.1

Move ${\sqrt{3}}^{- \frac{1}{3}}$ .

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

Step 2.3.1.6.4.2

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

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Step 2.3.1.6.4.2.1

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

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

Step 2.3.1.6.4.2.2

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

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

Step 2.3.1.6.4.3

Write $1$ as a fraction with a common denominator.

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

Step 2.3.1.6.4.4

Combine the numerators over the common denominator.

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

Step 2.3.1.6.4.5

Add $- 1$ and $3$ .

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

Step 2.3.1.6.5

Rewrite ${\sqrt{3}}^{\frac{2}{3}}$ as $\sqrt[3]{3}$ .

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Step 2.3.1.6.5.1

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

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

Step 2.3.1.6.5.2

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

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

Step 2.3.1.6.5.3

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

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

Step 2.3.1.6.5.4

Multiply $2$ by $3$ .

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

Step 2.3.1.6.5.5

Cancel the common factor of $2$ and $6$ .

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Step 2.3.1.6.5.5.1

Factor $2$ out of $2$ .

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

Step 2.3.1.6.5.5.2

Cancel the common factors.

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Step 2.3.1.6.5.5.2.1

Factor $2$ out of $6$ .

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

Step 2.3.1.6.5.5.2.2

Cancel the common factor.

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

Step 2.3.1.6.5.5.2.3

Rewrite the expression.

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

Step 2.3.1.6.5.6

Rewrite $3^{\frac{1}{3}}$ as $\sqrt[3]{3}$ .

$y - 1 = - \frac{x}{{(- 3)}^{\frac{1}{3}} {\sqrt{3}}^{\frac{1}{3}}} + {(- 3)}^{\frac{2}{3}} \sqrt[3]{3}$

Step 2.3.2

Add $1$ to both sides of the equation.

$y = - \frac{x}{{(- 3)}^{\frac{1}{3}} {\sqrt{3}}^{\frac{1}{3}}} + {(- 3)}^{\frac{2}{3}} \sqrt[3]{3} + 1$

Step 2.3.3

Write in $y = m x + b$ form.

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{1}{{(- 3)}^{\frac{1}{3}} {\sqrt{3}}^{\frac{1}{3}}} x) + {(- 3)}^{\frac{2}{3}} \sqrt[3]{3} + 1$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{1}{{(- 3)}^{\frac{1}{3}} {\sqrt{3}}^{\frac{1}{3}}} x + {(- 3)}^{\frac{2}{3}} \sqrt[3]{3} + 1$

Step 3