Trigonometry Examples

$\log (y^{2} - 1) - 3 \log (x) = - 2 \log (y + 1) + \log (9 x + x y)$

Step 1

Reorder $9 x$ and $x y$ .

$\log (y^{2} - 1) - 3 \log (x) = - 2 \log (y + 1) + \log (x y + 9 x)$

Step 2

Simplify the left side.

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Step 2.1

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

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Step 2.1.1

Simplify $- 3 \log (x)$ by moving $3$ inside the logarithm.

$\log (y^{2} - 1) - \log (x^{3}) = - 2 \log (y + 1) + \log (x y + 9 x)$

Step 2.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{y^{2} - 1}{x^{3}}) = - 2 \log (y + 1) + \log (x y + 9 x)$

Step 2.1.3

Simplify the numerator.

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Step 2.1.3.1

Rewrite $1$ as $1^{2}$ .

$\log (\frac{y^{2} - 1^{2}}{x^{3}}) = - 2 \log (y + 1) + \log (x y + 9 x)$

Step 2.1.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 1$ .

$\log (\frac{(y + 1) (y - 1)}{x^{3}}) = - 2 \log (y + 1) + \log (x y + 9 x)$

Step 3

Simplify the right side.

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Step 3.1

Simplify $- 2 \log (y + 1)$ by moving $2$ inside the logarithm.

$\log (\frac{(y + 1) (y - 1)}{x^{3}}) = - \log ({(y + 1)}^{2}) + \log (x y + 9 x)$

Step 4

Move all the terms containing a logarithm to the left side of the equation.

$\log (\frac{(y + 1) (y - 1)}{x^{3}}) + \log ({(y + 1)}^{2}) - \log (x y + 9 x) = 0$

Step 5

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (\frac{(y + 1) (y - 1)}{x^{3}} \cdot {(y + 1)}^{2}) - \log (x y + 9 x) = 0$

Step 6

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{\frac{(y + 1) (y - 1)}{x^{3}} \cdot {(y + 1)}^{2}}{x y + 9 x}) = 0$

Step 7

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

$\log (\frac{\frac{(y + 1) (y - 1) {(y + 1)}^{2}}{x^{3}}}{x y + 9 x}) = 0$

Step 8

Factor $x$ out of $x y + 9 x$ .

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Step 8.1

Factor $x$ out of $x y$ .

$\log (\frac{\frac{(y + 1) (y - 1) {(y + 1)}^{2}}{x^{3}}}{x (y) + 9 x}) = 0$

Step 8.2

Factor $x$ out of $9 x$ .

$\log (\frac{\frac{(y + 1) (y - 1) {(y + 1)}^{2}}{x^{3}}}{x (y) + x \cdot 9}) = 0$

Step 8.3

Factor $x$ out of $x (y) + x \cdot 9$ .

$\log (\frac{\frac{(y + 1) (y - 1) {(y + 1)}^{2}}{x^{3}}}{x (y + 9)}) = 0$

Step 9

Multiply $y + 1$ by ${(y + 1)}^{2}$ by adding the exponents.

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Step 9.1

Move ${(y + 1)}^{2}$ .

$\log (\frac{\frac{{(y + 1)}^{2} (y + 1) (y - 1)}{x^{3}}}{x (y + 9)}) = 0$

Step 9.2

Multiply ${(y + 1)}^{2}$ by $y + 1$ .

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Step 9.2.1

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

$\log (\frac{\frac{{(y + 1)}^{2} (y + 1) (y - 1)}{x^{3}}}{x (y + 9)}) = 0$

Step 9.2.2

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

$\log (\frac{\frac{{(y + 1)}^{2 + 1} (y - 1)}{x^{3}}}{x (y + 9)}) = 0$

Step 9.3

Add $2$ and $1$ .

$\log (\frac{\frac{{(y + 1)}^{3} (y - 1)}{x^{3}}}{x (y + 9)}) = 0$

Step 10

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{{(y + 1)}^{3} (y - 1)}{x^{3}} \cdot \frac{1}{x (y + 9)}) = 0$

Step 11

Combine.

$\log (\frac{{(y + 1)}^{3} (y - 1) \cdot 1}{x^{3} (x (y + 9))}) = 0$

Step 12

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

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Step 12.1

Move $x$ .

$\log (\frac{{(y + 1)}^{3} (y - 1) \cdot 1}{x \cdot x^{3} (y + 9)}) = 0$

Step 12.2

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

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Step 12.2.1

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

$\log (\frac{{(y + 1)}^{3} (y - 1) \cdot 1}{x \cdot x^{3} (y + 9)}) = 0$

Step 12.2.2

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

$\log (\frac{{(y + 1)}^{3} (y - 1) \cdot 1}{x^{1 + 3} (y + 9)}) = 0$

Step 12.3

Add $1$ and $3$ .

$\log (\frac{{(y + 1)}^{3} (y - 1) \cdot 1}{x^{4} (y + 9)}) = 0$

Step 13

Multiply ${(y + 1)}^{3} (y - 1)$ by $1$ .

$\log (\frac{{(y + 1)}^{3} (y - 1)}{x^{4} (y + 9)}) = 0$

Step 14

Rewrite $\log (\frac{{(y + 1)}^{3} (y - 1)}{x^{4} (y + 9)}) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$10^{0} = \frac{{(y + 1)}^{3} (y - 1)}{x^{4} (y + 9)}$

Step 15

Solve for $x$ .

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Step 15.1

Rewrite the equation as $\frac{{(y + 1)}^{3} (y - 1)}{x^{4} (y + 9)} = 10^{0}$ .

$\frac{{(y + 1)}^{3} (y - 1)}{x^{4} (y + 9)} = 10^{0}$

Step 15.2

Anything raised to $0$ is $1$ .

$\frac{{(y + 1)}^{3} (y - 1)}{x^{4} (y + 9)} = 1$

Step 15.3

Find the LCD of the terms in the equation.

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Step 15.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{4} (y + 9), 1$

Step 15.3.2

The LCM of one and any expression is the expression.

$x^{4} (y + 9)$

Step 15.4

Multiply each term in $\frac{{(y + 1)}^{3} (y - 1)}{x^{4} (y + 9)} = 1$ by $x^{4} (y + 9)$ to eliminate the fractions.

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Step 15.4.1

Multiply each term in $\frac{{(y + 1)}^{3} (y - 1)}{x^{4} (y + 9)} = 1$ by $x^{4} (y + 9)$ .

$\frac{{(y + 1)}^{3} (y - 1)}{x^{4} (y + 9)} (x^{4} (y + 9)) = 1 (x^{4} (y + 9))$

Step 15.4.2

Simplify the left side.

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Step 15.4.2.1

Simplify terms.

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Step 15.4.2.1.1

Cancel the common factor of $x^{4} (y + 9)$ .

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Step 15.4.2.1.1.1

Cancel the common factor.

$\frac{{(y + 1)}^{3} (y - 1)}{x^{4} (y + 9)} (x^{4} (y + 9)) = 1 (x^{4} (y + 9))$

Step 15.4.2.1.1.2

Rewrite the expression.

${(y + 1)}^{3} (y - 1) = 1 (x^{4} (y + 9))$

Step 15.4.2.1.2

Apply the distributive property.

${(y + 1)}^{3} y + {(y + 1)}^{3} \cdot - 1 = 1 (x^{4} (y + 9))$

Step 15.4.2.1.3

Move $- 1$ to the left of ${(y + 1)}^{3}$ .

${(y + 1)}^{3} y - 1 \cdot {(y + 1)}^{3} = 1 (x^{4} (y + 9))$

Step 15.4.2.2

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

${(y + 1)}^{3} y - {(y + 1)}^{3} = 1 (x^{4} (y + 9))$

Step 15.4.2.3

Reorder factors in ${(y + 1)}^{3} y - {(y + 1)}^{3}$ .

$y {(y + 1)}^{3} - {(y + 1)}^{3} = 1 (x^{4} (y + 9))$

Step 15.4.3

Simplify the right side.

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Step 15.4.3.1

Multiply $x^{4} (y + 9)$ by $1$ .

$y {(y + 1)}^{3} - {(y + 1)}^{3} = x^{4} (y + 9)$

Step 15.4.3.2

Apply the distributive property.

$y {(y + 1)}^{3} - {(y + 1)}^{3} = x^{4} y + x^{4} \cdot 9$

Step 15.4.3.3

Move $9$ to the left of $x^{4}$ .

$y {(y + 1)}^{3} - {(y + 1)}^{3} = x^{4} y + 9 x^{4}$

Step 15.5

Solve the equation.

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Step 15.5.1

Rewrite the equation as $x^{4} y + 9 x^{4} = y {(y + 1)}^{3} - {(y + 1)}^{3}$ .

$x^{4} y + 9 x^{4} = y {(y + 1)}^{3} - {(y + 1)}^{3}$

Step 15.5.2

Simplify $y {(y + 1)}^{3} - {(y + 1)}^{3}$ .

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Step 15.5.2.1

Simplify each term.

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Step 15.5.2.1.1

Use the Binomial Theorem.

$x^{4} y + 9 x^{4} = y (y^{3} + 3 y^{2} \cdot 1 + 3 y \cdot 1^{2} + 1^{3}) - {(y + 1)}^{3}$

Step 15.5.2.1.2

Simplify each term.

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Step 15.5.2.1.2.1

Multiply $3$ by $1$ .

$x^{4} y + 9 x^{4} = y (y^{3} + 3 y^{2} + 3 y \cdot 1^{2} + 1^{3}) - {(y + 1)}^{3}$

Step 15.5.2.1.2.2

One to any power is one.

$x^{4} y + 9 x^{4} = y (y^{3} + 3 y^{2} + 3 y \cdot 1 + 1^{3}) - {(y + 1)}^{3}$

Step 15.5.2.1.2.3

Multiply $3$ by $1$ .

$x^{4} y + 9 x^{4} = y (y^{3} + 3 y^{2} + 3 y + 1^{3}) - {(y + 1)}^{3}$

Step 15.5.2.1.2.4

One to any power is one.

$x^{4} y + 9 x^{4} = y (y^{3} + 3 y^{2} + 3 y + 1) - {(y + 1)}^{3}$

Step 15.5.2.1.3

Apply the distributive property.

$x^{4} y + 9 x^{4} = y \cdot y^{3} + y (3 y^{2}) + y (3 y) + y \cdot 1 - {(y + 1)}^{3}$

Step 15.5.2.1.4

Simplify.

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Step 15.5.2.1.4.1

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

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Step 15.5.2.1.4.1.1

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

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Step 15.5.2.1.4.1.1.1

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

$x^{4} y + 9 x^{4} = y^{1} y^{3} + y (3 y^{2}) + y (3 y) + y \cdot 1 - {(y + 1)}^{3}$

Step 15.5.2.1.4.1.1.2

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

$x^{4} y + 9 x^{4} = y^{1 + 3} + y (3 y^{2}) + y (3 y) + y \cdot 1 - {(y + 1)}^{3}$

Step 15.5.2.1.4.1.2

Add $1$ and $3$ .

$x^{4} y + 9 x^{4} = y^{4} + y (3 y^{2}) + y (3 y) + y \cdot 1 - {(y + 1)}^{3}$

Step 15.5.2.1.4.2

Rewrite using the commutative property of multiplication.

$x^{4} y + 9 x^{4} = y^{4} + 3 y \cdot y^{2} + y (3 y) + y \cdot 1 - {(y + 1)}^{3}$

Step 15.5.2.1.4.3

Rewrite using the commutative property of multiplication.

$x^{4} y + 9 x^{4} = y^{4} + 3 y \cdot y^{2} + 3 y \cdot y + y \cdot 1 - {(y + 1)}^{3}$

Step 15.5.2.1.4.4

Multiply $y$ by $1$ .

$x^{4} y + 9 x^{4} = y^{4} + 3 y \cdot y^{2} + 3 y \cdot y + y - {(y + 1)}^{3}$

Step 15.5.2.1.5

Simplify each term.

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Step 15.5.2.1.5.1

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

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Step 15.5.2.1.5.1.1

Move $y^{2}$ .

$x^{4} y + 9 x^{4} = y^{4} + 3 (y^{2} y) + 3 y \cdot y + y - {(y + 1)}^{3}$

Step 15.5.2.1.5.1.2

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

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Step 15.5.2.1.5.1.2.1

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

$x^{4} y + 9 x^{4} = y^{4} + 3 (y^{2} y^{1}) + 3 y \cdot y + y - {(y + 1)}^{3}$

Step 15.5.2.1.5.1.2.2

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

$x^{4} y + 9 x^{4} = y^{4} + 3 y^{2 + 1} + 3 y \cdot y + y - {(y + 1)}^{3}$

Step 15.5.2.1.5.1.3

Add $2$ and $1$ .

$x^{4} y + 9 x^{4} = y^{4} + 3 y^{3} + 3 y \cdot y + y - {(y + 1)}^{3}$

Step 15.5.2.1.5.2

Multiply $y$ by $y$ by adding the exponents.

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Step 15.5.2.1.5.2.1

Move $y$ .

$x^{4} y + 9 x^{4} = y^{4} + 3 y^{3} + 3 (y \cdot y) + y - {(y + 1)}^{3}$

Step 15.5.2.1.5.2.2

Multiply $y$ by $y$ .

$x^{4} y + 9 x^{4} = y^{4} + 3 y^{3} + 3 y^{2} + y - {(y + 1)}^{3}$

Step 15.5.2.1.6

Use the Binomial Theorem.

$x^{4} y + 9 x^{4} = y^{4} + 3 y^{3} + 3 y^{2} + y - (y^{3} + 3 y^{2} \cdot 1 + 3 y \cdot 1^{2} + 1^{3})$

Step 15.5.2.1.7

Simplify each term.

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Step 15.5.2.1.7.1

Multiply $3$ by $1$ .

$x^{4} y + 9 x^{4} = y^{4} + 3 y^{3} + 3 y^{2} + y - (y^{3} + 3 y^{2} + 3 y \cdot 1^{2} + 1^{3})$

Step 15.5.2.1.7.2

One to any power is one.

$x^{4} y + 9 x^{4} = y^{4} + 3 y^{3} + 3 y^{2} + y - (y^{3} + 3 y^{2} + 3 y \cdot 1 + 1^{3})$

Step 15.5.2.1.7.3

Multiply $3$ by $1$ .

$x^{4} y + 9 x^{4} = y^{4} + 3 y^{3} + 3 y^{2} + y - (y^{3} + 3 y^{2} + 3 y + 1^{3})$

Step 15.5.2.1.7.4

One to any power is one.

$x^{4} y + 9 x^{4} = y^{4} + 3 y^{3} + 3 y^{2} + y - (y^{3} + 3 y^{2} + 3 y + 1)$

Step 15.5.2.1.8

Apply the distributive property.

$x^{4} y + 9 x^{4} = y^{4} + 3 y^{3} + 3 y^{2} + y - y^{3} - (3 y^{2}) - (3 y) - 1 \cdot 1$

Step 15.5.2.1.9

Simplify.

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Step 15.5.2.1.9.1

Multiply $3$ by $- 1$ .

$x^{4} y + 9 x^{4} = y^{4} + 3 y^{3} + 3 y^{2} + y - y^{3} - 3 y^{2} - (3 y) - 1 \cdot 1$

Step 15.5.2.1.9.2

Multiply $3$ by $- 1$ .

$x^{4} y + 9 x^{4} = y^{4} + 3 y^{3} + 3 y^{2} + y - y^{3} - 3 y^{2} - 3 y - 1 \cdot 1$

Step 15.5.2.1.9.3

Multiply $- 1$ by $1$ .

$x^{4} y + 9 x^{4} = y^{4} + 3 y^{3} + 3 y^{2} + y - y^{3} - 3 y^{2} - 3 y - 1$

Step 15.5.2.2

Simplify by adding terms.

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Step 15.5.2.2.1

Combine the opposite terms in $y^{4} + 3 y^{3} + 3 y^{2} + y - y^{3} - 3 y^{2} - 3 y - 1$ .

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Step 15.5.2.2.1.1

Subtract $3 y^{2}$ from $3 y^{2}$ .

$x^{4} y + 9 x^{4} = y^{4} + 3 y^{3} + y - y^{3} + 0 - 3 y - 1$

Step 15.5.2.2.1.2

Add $y^{4} + 3 y^{3} + y - y^{3}$ and $0$ .

$x^{4} y + 9 x^{4} = y^{4} + 3 y^{3} + y - y^{3} - 3 y - 1$

Step 15.5.2.2.2

Subtract $y^{3}$ from $3 y^{3}$ .

$x^{4} y + 9 x^{4} = y^{4} + 2 y^{3} + y - 3 y - 1$

Step 15.5.2.2.3

Subtract $3 y$ from $y$ .

$x^{4} y + 9 x^{4} = y^{4} + 2 y^{3} - 2 y - 1$

Step 15.5.3

Factor $x^{4}$ out of $x^{4} y + 9 x^{4}$ .

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Step 15.5.3.1

Factor $x^{4}$ out of $x^{4} y$ .

$x^{4} (y) + 9 x^{4} = y^{4} + 2 y^{3} - 2 y - 1$

Step 15.5.3.2

Factor $x^{4}$ out of $9 x^{4}$ .

$x^{4} (y) + x^{4} \cdot 9 = y^{4} + 2 y^{3} - 2 y - 1$

Step 15.5.3.3

Factor $x^{4}$ out of $x^{4} (y) + x^{4} \cdot 9$ .

$x^{4} (y + 9) = y^{4} + 2 y^{3} - 2 y - 1$

Step 15.5.4

Divide each term in $x^{4} (y + 9) = y^{4} + 2 y^{3} - 2 y - 1$ by $y + 9$ and simplify.

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Step 15.5.4.1

Divide each term in $x^{4} (y + 9) = y^{4} + 2 y^{3} - 2 y - 1$ by $y + 9$ .

$\frac{x^{4} (y + 9)}{y + 9} = \frac{y^{4}}{y + 9} + \frac{2 y^{3}}{y + 9} + \frac{- 2 y}{y + 9} + \frac{- 1}{y + 9}$

Step 15.5.4.2

Simplify the left side.

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Step 15.5.4.2.1

Cancel the common factor of $y + 9$ .

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Step 15.5.4.2.1.1

Cancel the common factor.

$\frac{x^{4} (y + 9)}{y + 9} = \frac{y^{4}}{y + 9} + \frac{2 y^{3}}{y + 9} + \frac{- 2 y}{y + 9} + \frac{- 1}{y + 9}$

Step 15.5.4.2.1.2

Divide $x^{4}$ by $1$ .

$x^{4} = \frac{y^{4}}{y + 9} + \frac{2 y^{3}}{y + 9} + \frac{- 2 y}{y + 9} + \frac{- 1}{y + 9}$

Step 15.5.4.3

Simplify the right side.

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Step 15.5.4.3.1

Combine the numerators over the common denominator.

$x^{4} = \frac{y^{4} + 2 y^{3} - 2 y - 1}{y + 9}$

Step 15.5.5

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

$x = \pm \sqrt[4]{\frac{y^{4} + 2 y^{3} - 2 y - 1}{y + 9}}$

Step 15.5.6

Simplify $\pm \sqrt[4]{\frac{y^{4} + 2 y^{3} - 2 y - 1}{y + 9}}$ .

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Step 15.5.6.1

Rewrite $\sqrt[4]{\frac{y^{4} + 2 y^{3} - 2 y - 1}{y + 9}}$ as $\frac{\sqrt[4]{y^{4} + 2 y^{3} - 2 y - 1}}{\sqrt[4]{y + 9}}$ .

$x = \pm \frac{\sqrt[4]{y^{4} + 2 y^{3} - 2 y - 1}}{\sqrt[4]{y + 9}}$

Step 15.5.6.2

Multiply $\frac{\sqrt[4]{y^{4} + 2 y^{3} - 2 y - 1}}{\sqrt[4]{y + 9}}$ by $\frac{{\sqrt[4]{y + 9}}^{3}}{{\sqrt[4]{y + 9}}^{3}}$ .

$x = \pm \frac{\sqrt[4]{y^{4} + 2 y^{3} - 2 y - 1}}{\sqrt[4]{y + 9}} \cdot \frac{{\sqrt[4]{y + 9}}^{3}}{{\sqrt[4]{y + 9}}^{3}}$

Step 15.5.6.3

Combine and simplify the denominator.

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Step 15.5.6.3.1

Multiply $\frac{\sqrt[4]{y^{4} + 2 y^{3} - 2 y - 1}}{\sqrt[4]{y + 9}}$ by $\frac{{\sqrt[4]{y + 9}}^{3}}{{\sqrt[4]{y + 9}}^{3}}$ .

$x = \pm \frac{\sqrt[4]{y^{4} + 2 y^{3} - 2 y - 1} {\sqrt[4]{y + 9}}^{3}}{\sqrt[4]{y + 9} {\sqrt[4]{y + 9}}^{3}}$

Step 15.5.6.3.2

Raise $\sqrt[4]{y + 9}$ to the power of $1$ .

$x = \pm \frac{\sqrt[4]{y^{4} + 2 y^{3} - 2 y - 1} {\sqrt[4]{y + 9}}^{3}}{{\sqrt[4]{y + 9}}^{1} {\sqrt[4]{y + 9}}^{3}}$

Step 15.5.6.3.3

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

$x = \pm \frac{\sqrt[4]{y^{4} + 2 y^{3} - 2 y - 1} {\sqrt[4]{y + 9}}^{3}}{{\sqrt[4]{y + 9}}^{1 + 3}}$

Step 15.5.6.3.4

Add $1$ and $3$ .

$x = \pm \frac{\sqrt[4]{y^{4} + 2 y^{3} - 2 y - 1} {\sqrt[4]{y + 9}}^{3}}{{\sqrt[4]{y + 9}}^{4}}$

Step 15.5.6.3.5

Rewrite ${\sqrt[4]{y + 9}}^{4}$ as $y + 9$ .

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Step 15.5.6.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{y + 9}$ as ${(y + 9)}^{\frac{1}{4}}$ .

$x = \pm \frac{\sqrt[4]{y^{4} + 2 y^{3} - 2 y - 1} {\sqrt[4]{y + 9}}^{3}}{{({(y + 9)}^{\frac{1}{4}})}^{4}}$

Step 15.5.6.3.5.2

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

$x = \pm \frac{\sqrt[4]{y^{4} + 2 y^{3} - 2 y - 1} {\sqrt[4]{y + 9}}^{3}}{{(y + 9)}^{\frac{1}{4} \cdot 4}}$

Step 15.5.6.3.5.3

Combine $\frac{1}{4}$ and $4$ .

$x = \pm \frac{\sqrt[4]{y^{4} + 2 y^{3} - 2 y - 1} {\sqrt[4]{y + 9}}^{3}}{{(y + 9)}^{\frac{4}{4}}}$

Step 15.5.6.3.5.4

Cancel the common factor of $4$ .

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Step 15.5.6.3.5.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt[4]{y^{4} + 2 y^{3} - 2 y - 1} {\sqrt[4]{y + 9}}^{3}}{{(y + 9)}^{\frac{4}{4}}}$

Step 15.5.6.3.5.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt[4]{y^{4} + 2 y^{3} - 2 y - 1} {\sqrt[4]{y + 9}}^{3}}{{(y + 9)}^{1}}$

Step 15.5.6.3.5.5

Simplify.

$x = \pm \frac{\sqrt[4]{y^{4} + 2 y^{3} - 2 y - 1} {\sqrt[4]{y + 9}}^{3}}{y + 9}$

Step 15.5.6.4

Rewrite ${\sqrt[4]{y + 9}}^{3}$ as $\sqrt[4]{{(y + 9)}^{3}}$ .

$x = \pm \frac{\sqrt[4]{y^{4} + 2 y^{3} - 2 y - 1} \sqrt[4]{{(y + 9)}^{3}}}{y + 9}$

Step 15.5.6.5

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt[4]{(y^{4} + 2 y^{3} - 2 y - 1) {(y + 9)}^{3}}}{y + 9}$

Step 15.5.7

The complete solution is the result of both the positive and negative portions of the solution.

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Step 15.5.7.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt[4]{(y^{4} + 2 y^{3} - 2 y - 1) {(y + 9)}^{3}}}{y + 9}$

Step 15.5.7.2

Next, use the negative value of the $\pm$ to find the second solution.