Trigonometry Examples

${(\sqrt[3]{64 x^{6}})}^{5}$

Step 1

Interchange the variables.

$x = {(\sqrt[3]{64 y^{6}})}^{5}$

Step 2

Solve for $y$ .

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Step 2.1

Simplify ${\sqrt[3]{64 y^{6}}}^{5}$ .

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Step 2.1.1

Rewrite $64 y^{6}$ as ${(4 y^{2})}^{3}$ .

$x = {\sqrt[3]{{(4 y^{2})}^{3}}}^{5}$

Step 2.1.2

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

$x = {(4 y^{2})}^{5}$

Step 2.1.3

Apply the product rule to $4 y^{2}$ .

$x = 4^{5} {(y^{2})}^{5}$

Step 2.1.4

Raise $4$ to the power of $5$ .

$x = 1024 {(y^{2})}^{5}$

Step 2.1.5

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

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Step 2.1.5.1

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

$x = 1024 y^{2 \cdot 5}$

Step 2.1.5.2

Multiply $2$ by $5$ .

$x = 1024 y^{10}$

Step 2.2

Rewrite the equation as $1024 y^{10} = x$ .

$1024 y^{10} = x$

Step 2.3

Divide each term in $1024 y^{10} = x$ by $1024$ and simplify.

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Step 2.3.1

Divide each term in $1024 y^{10} = x$ by $1024$ .

$\frac{1024 y^{10}}{1024} = \frac{x}{1024}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $1024$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{1024 y^{10}}{1024} = \frac{x}{1024}$

Step 2.3.2.1.2

Divide $y^{10}$ by $1$ .

$y^{10} = \frac{x}{1024}$

Step 2.4

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

$y = \pm \sqrt[10]{\frac{x}{1024}}$

Step 2.5

Simplify $\pm \sqrt[10]{\frac{x}{1024}}$ .

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Step 2.5.1

Rewrite $\frac{x}{1024}$ as ${(\frac{1}{2})}^{10} x$ .

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Step 2.5.1.1

Factor the perfect power $1^{10}$ out of $x$ .

$y = \pm \sqrt[10]{\frac{1^{10} x}{1024}}$

Step 2.5.1.2

Factor the perfect power $2^{10}$ out of $1024$ .

$y = \pm \sqrt[10]{\frac{1^{10} x}{2^{10} \cdot 1}}$

Step 2.5.1.3

Rearrange the fraction $\frac{1^{10} x}{2^{10} \cdot 1}$ .

$y = \pm \sqrt[10]{{(\frac{1}{2})}^{10} x}$

Step 2.5.2

Pull terms out from under the radical.

$y = \pm \frac{1}{2} \sqrt[10]{x}$

Step 2.5.3

Combine $\frac{1}{2}$ and $\sqrt[10]{x}$ .

$y = \pm \frac{\sqrt[10]{x}}{2}$

Step 2.6

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

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Step 2.6.1

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

$y = \frac{\sqrt[10]{x}}{2}$

Step 2.6.2

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

$y = - \frac{\sqrt[10]{x}}{2}$

Step 2.6.3

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

$y = \frac{\sqrt[10]{x}}{2}$

$y = - \frac{\sqrt[10]{x}}{2}$

$y = \frac{\sqrt[10]{x}}{2}$

$y = - \frac{\sqrt[10]{x}}{2}$

$y = \frac{\sqrt[10]{x}}{2}$

$y = - \frac{\sqrt[10]{x}}{2}$

Step 3

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

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

Step 4

Verify if $f^{- 1} (x) = \frac{\sqrt[10]{x}}{2}, - \frac{\sqrt[10]{x}}{2}$ is the inverse of $f (x) = {(\sqrt[3]{64 x^{6}})}^{5}$ .

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Step 4.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = {(\sqrt[3]{64 x^{6}})}^{5}$ and $f^{- 1} (x) = \frac{\sqrt[10]{x}}{2}, - \frac{\sqrt[10]{x}}{2}$ and compare them.

Step 4.2

Find the domain of $\frac{\sqrt[10]{x}}{2}$ .

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Step 4.2.1

Set the radicand in $\sqrt[10]{x}$ greater than or equal to $0$ to find where the expression is defined.

$x \geq 0$

Step 4.2.2

The domain is all values of $x$ that make the expression defined.

$[0, \infty)$

Step 4.3

Since the domain of $f^{- 1} (x) = \frac{\sqrt[10]{x}}{2}, - \frac{\sqrt[10]{x}}{2}$ is not equal to the range of $f (x) = {(\sqrt[3]{64 x^{6}})}^{5}$ , then $f^{- 1} (x) = \frac{\sqrt[10]{x}}{2}, - \frac{\sqrt[10]{x}}{2}$ is not an inverse of $f (x) = {(\sqrt[3]{64 x^{6}})}^{5}$ .

There is no inverse

Step 5