Trigonometry Examples

$\sqrt[5]{2 x^{2}}$

Step 1

Interchange the variables.

$x = \sqrt[5]{2 y^{2}}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\sqrt[5]{2 y^{2}} = x$ .

$\sqrt[5]{2 y^{2}} = x$

Step 2.2

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $5$ .

${\sqrt[5]{2 y^{2}}}^{5} = x^{5}$

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 $\sqrt[5]{2 y^{2}}$ as ${(2 y^{2})}^{\frac{1}{5}}$ .

${({(2 y^{2})}^{\frac{1}{5}})}^{5} = x^{5}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify ${({(2 y^{2})}^{\frac{1}{5}})}^{5}$ .

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Step 2.3.2.1.1

Multiply the exponents in ${({(2 y^{2})}^{\frac{1}{5}})}^{5}$ .

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Step 2.3.2.1.1.1

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

${(2 y^{2})}^{\frac{1}{5} \cdot 5} = x^{5}$

Step 2.3.2.1.1.2

Cancel the common factor of $5$ .

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Step 2.3.2.1.1.2.1

Cancel the common factor.

${(2 y^{2})}^{\frac{1}{5} \cdot 5} = x^{5}$

Step 2.3.2.1.1.2.2

Rewrite the expression.

${(2 y^{2})}^{1} = x^{5}$

Step 2.3.2.1.2

Simplify.

$2 y^{2} = x^{5}$

Step 2.4

Solve for $y$ .

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Step 2.4.1

Divide each term in $2 y^{2} = x^{5}$ by $2$ and simplify.

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Step 2.4.1.1

Divide each term in $2 y^{2} = x^{5}$ by $2$ .

$\frac{2 y^{2}}{2} = \frac{x^{5}}{2}$

Step 2.4.1.2

Simplify the left side.

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Step 2.4.1.2.1

Cancel the common factor of $2$ .

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Step 2.4.1.2.1.1

Cancel the common factor.

$\frac{2 y^{2}}{2} = \frac{x^{5}}{2}$

Step 2.4.1.2.1.2

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

$y^{2} = \frac{x^{5}}{2}$

Step 2.4.2

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

$y = \pm \sqrt{\frac{x^{5}}{2}}$

Step 2.4.3

Simplify $\pm \sqrt{\frac{x^{5}}{2}}$ .

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Step 2.4.3.1

Rewrite $\sqrt{\frac{x^{5}}{2}}$ as $\frac{\sqrt{x^{5}}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{x^{5}}}{\sqrt{2}}$

Step 2.4.3.2

Simplify the numerator.

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Step 2.4.3.2.1

Rewrite $x^{5}$ as ${(x^{2})}^{2} x$ .

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Step 2.4.3.2.1.1

Factor out $x^{4}$ .

$y = \pm \frac{\sqrt{x^{4} x}}{\sqrt{2}}$

Step 2.4.3.2.1.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 2.4.3.2.2

Pull terms out from under the radical.

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

Step 2.4.3.3

Multiply $\frac{x^{2} \sqrt{x}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 2.4.3.4

Combine and simplify the denominator.

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Step 2.4.3.4.1

Multiply $\frac{x^{2} \sqrt{x}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 2.4.3.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$y = \pm \frac{x^{2} \sqrt{x} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 2.4.3.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$y = \pm \frac{x^{2} \sqrt{x} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 2.4.3.4.4

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

$y = \pm \frac{x^{2} \sqrt{x} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 2.4.3.4.5

Add $1$ and $1$ .

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

Step 2.4.3.4.6

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

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Step 2.4.3.4.6.1

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

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

Step 2.4.3.4.6.2

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

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

Step 2.4.3.4.6.3

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

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

Step 2.4.3.4.6.4

Cancel the common factor of $2$ .

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Step 2.4.3.4.6.4.1

Cancel the common factor.

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

Step 2.4.3.4.6.4.2

Rewrite the expression.

$y = \pm \frac{x^{2} \sqrt{x} \sqrt{2}}{2^{1}}$

Step 2.4.3.4.6.5

Evaluate the exponent.

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

Step 2.4.3.5

Combine using the product rule for radicals.

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

Step 2.4.4

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

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Step 2.4.4.1

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