Trigonometry Examples

$\sqrt{w^{11}}$

Step 1

Interchange the variables.

$w = \sqrt{y^{11}}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\sqrt{y^{11}} = w$ .

$\sqrt{y^{11}} = w$

Step 2.2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{y^{11}}}^{2} = w^{2}$

Step 2.3

Simplify each side of the equation.

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Step 2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y^{11}}$ as $y^{\frac{11}{2}}$ .

${(y^{\frac{11}{2}})}^{2} = w^{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Multiply the exponents in ${(y^{\frac{11}{2}})}^{2}$ .

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Step 2.3.2.1.1

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

$y^{\frac{11}{2} \cdot 2} = w^{2}$

Step 2.3.2.1.2

Cancel the common factor of $2$ .

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Step 2.3.2.1.2.1

Cancel the common factor.

$y^{\frac{11}{2} \cdot 2} = w^{2}$

Step 2.3.2.1.2.2

Rewrite the expression.

$y^{11} = w^{2}$

Step 2.4

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

$y = \sqrt[11]{w^{2}}$

Step 3

Replace $y$ with $f^{- 1} (w)$ to show the final answer.

$f^{- 1} (w) = \sqrt[11]{w^{2}}$

Step 4

Verify if $f^{- 1} (w) = \sqrt[11]{w^{2}}$ is the inverse of $f (w) = \sqrt{w^{11}}$ .

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Step 4.1

To verify the inverse, check if $f^{- 1} (f (w)) = w$ and $f (f^{- 1} (w)) = w$ .

Step 4.2

Evaluate $f^{- 1} (f (w))$ .

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Step 4.2.1

Set up the composite result function.

$f^{- 1} (f (w))$

Step 4.2.2

Evaluate $f^{- 1} (\sqrt{w^{11}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt{w^{11}}) = \sqrt[11]{{(\sqrt{w^{11}})}^{2}}$

Step 4.2.3

Rewrite $w^{11}$ as ${(w^{5})}^{2} w$ .

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Step 4.2.3.1

Factor out $w^{10}$ .

$f^{- 1} (\sqrt{w^{11}}) = \sqrt[11]{{\sqrt{w^{10} w}}^{2}}$

Step 4.2.3.2

Rewrite $w^{10}$ as ${(w^{5})}^{2}$ .

$f^{- 1} (\sqrt{w^{11}}) = \sqrt[11]{{\sqrt{{(w^{5})}^{2} w}}^{2}}$

Step 4.2.4

Pull terms out from under the radical.

$f^{- 1} (\sqrt{w^{11}}) = \sqrt[11]{{(w^{5} \sqrt{w})}^{2}}$

Step 4.2.5

Apply basic rules of exponents.

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Step 4.2.5.1

Apply the product rule to $w^{5} \sqrt{w}$ .

$f^{- 1} (\sqrt{w^{11}}) = \sqrt[11]{{(w^{5})}^{2} {\sqrt{w}}^{2}}$

Step 4.2.5.2

Multiply the exponents in ${(w^{5})}^{2}$ .

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Step 4.2.5.2.1

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

$f^{- 1} (\sqrt{w^{11}}) = \sqrt[11]{w^{5 \cdot 2} {\sqrt{w}}^{2}}$

Step 4.2.5.2.2

Multiply $5$ by $2$ .

$f^{- 1} (\sqrt{w^{11}}) = \sqrt[11]{w^{10} {\sqrt{w}}^{2}}$

Step 4.2.6

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

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Step 4.2.6.1

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

$f^{- 1} (\sqrt{w^{11}}) = \sqrt[11]{w^{10} {(w^{\frac{1}{2}})}^{2}}$

Step 4.2.6.2

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

$f^{- 1} (\sqrt{w^{11}}) = \sqrt[11]{w^{10} w^{\frac{1}{2} \cdot 2}}$

Step 4.2.6.3

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

$f^{- 1} (\sqrt{w^{11}}) = \sqrt[11]{w^{10} w^{\frac{2}{2}}}$

Step 4.2.6.4

Cancel the common factor of $2$ .

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Step 4.2.6.4.1

Cancel the common factor.

$f^{- 1} (\sqrt{w^{11}}) = \sqrt[11]{w^{10} w^{\frac{2}{2}}}$

Step 4.2.6.4.2

Rewrite the expression.

$f^{- 1} (\sqrt{w^{11}}) = \sqrt[11]{w^{10} w}$

Step 4.2.6.5

Simplify.

$f^{- 1} (\sqrt{w^{11}}) = \sqrt[11]{w^{10} w}$

Step 4.2.7

Multiply $w^{10}$ by $w$ by adding the exponents.

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Step 4.2.7.1

Multiply $w^{10}$ by $w$ .

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Step 4.2.7.1.1

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

$f^{- 1} (\sqrt{w^{11}}) = \sqrt[11]{w^{10} w}$

Step 4.2.7.1.2

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

$f^{- 1} (\sqrt{w^{11}}) = \sqrt[11]{w^{10 + 1}}$

Step 4.2.7.2

Add $10$ and $1$ .

$f^{- 1} (\sqrt{w^{11}}) = \sqrt[11]{w^{11}}$

Step 4.2.8

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

$f^{- 1} (\sqrt{w^{11}}) = w$

Step 4.3

Evaluate $f (f^{- 1} (w))$ .

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Step 4.3.1

Set up the composite result function.

$f (f^{- 1} (w))$

Step 4.3.2

Evaluate $f (\sqrt[11]{w^{2}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\sqrt[11]{w^{2}}) = \sqrt{{(\sqrt[11]{w^{2}})}^{11}}$

Step 4.3.3

Rewrite ${\sqrt[11]{w^{2}}}^{11}$ as $w^{2}$ .

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Step 4.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[11]{w^{2}}$ as $w^{\frac{2}{11}}$ .

$f (\sqrt[11]{w^{2}}) = \sqrt{{(w^{\frac{2}{11}})}^{11}}$

Step 4.3.3.2

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

$f (\sqrt[11]{w^{2}}) = \sqrt{w^{\frac{2}{11} \cdot 11}}$

Step 4.3.3.3

Combine $\frac{2}{11}$ and $11$ .

$f (\sqrt[11]{w^{2}}) = \sqrt{w^{\frac{2 \cdot 11}{11}}}$

Step 4.3.3.4

Multiply $2$ by $11$ .

$f (\sqrt[11]{w^{2}}) = \sqrt{w^{\frac{22}{11}}}$

Step 4.3.3.5

Cancel the common factor of $22$ and $11$ .

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Step 4.3.3.5.1

Factor $11$ out of $22$ .

$f (\sqrt[11]{w^{2}}) = \sqrt{w^{\frac{11 \cdot 2}{11}}}$

Step 4.3.3.5.2

Cancel the common factors.

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Step 4.3.3.5.2.1

Factor $11$ out of $11$ .

$f (\sqrt[11]{w^{2}}) = \sqrt{w^{\frac{11 \cdot 2}{11 (1)}}}$

Step 4.3.3.5.2.2

Cancel the common factor.

$f (\sqrt[11]{w^{2}}) = \sqrt{w^{\frac{11 \cdot 2}{11 \cdot 1}}}$

Step 4.3.3.5.2.3

Rewrite the expression.

$f (\sqrt[11]{w^{2}}) = \sqrt{w^{\frac{2}{1}}}$

Step 4.3.3.5.2.4

Divide $2$ by $1$ .

$f (\sqrt[11]{w^{2}}) = \sqrt{w^{2}}$

Step 4.3.4

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

$f (\sqrt[11]{w^{2}}) = w$

Step 4.4

Since $f^{- 1} (f (w)) = w$ and $f (f^{- 1} (w)) = w$ , then $f^{- 1} (w) = \sqrt[11]{w^{2}}$ is the inverse of $f (w) = \sqrt{w^{11}}$ .

$f^{- 1} (w) = \sqrt[11]{w^{2}}$