Algebra Examples

$x = 3 y^{4}$

Step 1

Rewrite the equation as $3 y^{4} = x$ .

$3 y^{4} = x$

Step 2

Divide each term in $3 y^{4} = x$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 y^{4} = x$ by $3$ .

$\frac{3 y^{4}}{3} = \frac{x}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 y^{4}}{3} = \frac{x}{3}$

Step 2.2.1.2

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

$y^{4} = \frac{x}{3}$

Step 3

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

$y = \pm \sqrt[4]{\frac{x}{3}}$

Step 4

Simplify $\pm \sqrt[4]{\frac{x}{3}}$ .

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Step 4.1

Rewrite $\sqrt[4]{\frac{x}{3}}$ as $\frac{\sqrt[4]{x}}{\sqrt[4]{3}}$ .

$y = \pm \frac{\sqrt[4]{x}}{\sqrt[4]{3}}$

Step 4.2

Multiply $\frac{\sqrt[4]{x}}{\sqrt[4]{3}}$ by $\frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$ .

$y = \pm \frac{\sqrt[4]{x}}{\sqrt[4]{3}} \cdot \frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

Multiply $\frac{\sqrt[4]{x}}{\sqrt[4]{3}}$ by $\frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$ .

$y = \pm \frac{\sqrt[4]{x} {\sqrt[4]{3}}^{3}}{\sqrt[4]{3} {\sqrt[4]{3}}^{3}}$

Step 4.3.2

Raise $\sqrt[4]{3}$ to the power of $1$ .

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

Step 4.3.3

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

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

Step 4.3.4

Add $1$ and $3$ .

$y = \pm \frac{\sqrt[4]{x} {\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{4}}$

Step 4.3.5

Rewrite ${\sqrt[4]{3}}^{4}$ as $3$ .

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Step 4.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3}$ as $3^{\frac{1}{4}}$ .

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

Step 4.3.5.2

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

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

Step 4.3.5.3

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

$y = \pm \frac{\sqrt[4]{x} {\sqrt[4]{3}}^{3}}{3^{\frac{4}{4}}}$

Step 4.3.5.4

Cancel the common factor of $4$ .

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Step 4.3.5.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt[4]{x} {\sqrt[4]{3}}^{3}}{3^{\frac{4}{4}}}$

Step 4.3.5.4.2

Rewrite the expression.

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

Step 4.3.5.5

Evaluate the exponent.

$y = \pm \frac{\sqrt[4]{x} {\sqrt[4]{3}}^{3}}{3}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Rewrite ${\sqrt[4]{3}}^{3}$ as $\sqrt[4]{3^{3}}$ .

$y = \pm \frac{\sqrt[4]{x} \sqrt[4]{3^{3}}}{3}$

Step 4.4.2

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

$y = \pm \frac{\sqrt[4]{x} \sqrt[4]{27}}{3}$

Step 4.5

Simplify with factoring out.

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Step 4.5.1

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt[4]{x \cdot 27}}{3}$

Step 4.5.2

Reorder factors in $\pm \frac{\sqrt[4]{x \cdot 27}}{3}$ .

$y = \pm \frac{\sqrt[4]{27 x}}{3}$

Step 5

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

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Step 5.1

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

$y = \frac{\sqrt[4]{27 x}}{3}$

Step 5.2

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

$y = - \frac{\sqrt[4]{27 x}}{3}$

Step 5.3

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

$y = \frac{\sqrt[4]{27 x}}{3}$

$y = - \frac{\sqrt[4]{27 x}}{3}$

$y = \frac{\sqrt[4]{27 x}}{3}$

$y = - \frac{\sqrt[4]{27 x}}{3}$

Step 6

To rewrite as a function of $x$ , write the equation so that $y$ is by itself on one side of the equal sign and an expression involving only $x$ is on the other side.

$f (x) = \frac{\sqrt[4]{27 x}}{3}, - \frac{\sqrt[4]{27 x}}{3}$