Algebra Examples

$\sqrt[5]{\frac{1}{x^{3}}}$

Step 1

Rewrite $\sqrt[5]{\frac{1}{x^{3}}}$ as $\frac{\sqrt[5]{1}}{\sqrt[5]{x^{3}}}$ .

$\frac{\sqrt[5]{1}}{\sqrt[5]{x^{3}}}$

Step 2

Any root of $1$ is $1$ .

$\frac{1}{\sqrt[5]{x^{3}}}$

Step 3

Multiply $\frac{1}{\sqrt[5]{x^{3}}}$ by $\frac{{\sqrt[5]{x^{3}}}^{4}}{{\sqrt[5]{x^{3}}}^{4}}$ .

$\frac{1}{\sqrt[5]{x^{3}}} \cdot \frac{{\sqrt[5]{x^{3}}}^{4}}{{\sqrt[5]{x^{3}}}^{4}}$

Step 4

Combine and simplify the denominator.

Tap for more steps...

Multiply $\frac{1}{\sqrt[5]{x^{3}}}$ by $\frac{{\sqrt[5]{x^{3}}}^{4}}{{\sqrt[5]{x^{3}}}^{4}}$ .

$\frac{{\sqrt[5]{x^{3}}}^{4}}{\sqrt[5]{x^{3}} {\sqrt[5]{x^{3}}}^{4}}$

Raise $\sqrt[5]{x^{3}}$ to the power of $1$ .

$\frac{{\sqrt[5]{x^{3}}}^{4}}{{\sqrt[5]{x^{3}}}^{1} {\sqrt[5]{x^{3}}}^{4}}$

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

$\frac{{\sqrt[5]{x^{3}}}^{4}}{{\sqrt[5]{x^{3}}}^{1 + 4}}$

Add $1$ and $4$ .

$\frac{{\sqrt[5]{x^{3}}}^{4}}{{\sqrt[5]{x^{3}}}^{5}}$

Rewrite ${\sqrt[5]{x^{3}}}^{5}$ as $x^{3}$ .

Tap for more steps...

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{3}}$ as $x^{\frac{3}{5}}$ .

$\frac{{\sqrt[5]{x^{3}}}^{4}}{{(x^{\frac{3}{5}})}^{5}}$

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

$\frac{{\sqrt[5]{x^{3}}}^{4}}{x^{\frac{3}{5} \cdot 5}}$

Combine $\frac{3}{5}$ and $5$ .

$\frac{{\sqrt[5]{x^{3}}}^{4}}{x^{\frac{3 \cdot 5}{5}}}$

Multiply $3$ by $5$ .

$\frac{{\sqrt[5]{x^{3}}}^{4}}{x^{\frac{15}{5}}}$

Cancel the common factor of $15$ and $5$ .

Tap for more steps...

Factor $5$ out of $15$ .

$\frac{{\sqrt[5]{x^{3}}}^{4}}{x^{\frac{5 \cdot 3}{5}}}$

Cancel the common factors.

Tap for more steps...

Factor $5$ out of $5$ .

$\frac{{\sqrt[5]{x^{3}}}^{4}}{x^{\frac{5 \cdot 3}{5 (1)}}}$

Cancel the common factor.

$\frac{{\sqrt[5]{x^{3}}}^{4}}{x^{\frac{5 \cdot 3}{5 \cdot 1}}}$

Rewrite the expression.

$\frac{{\sqrt[5]{x^{3}}}^{4}}{x^{\frac{3}{1}}}$

Divide $3$ by $1$ .

$\frac{{\sqrt[5]{x^{3}}}^{4}}{x^{3}}$

Step 5

Simplify the numerator.

Tap for more steps...

Rewrite ${\sqrt[5]{x^{3}}}^{4}$ as ${({(x^{3})}^{4})}^{\frac{1}{5}}$ .

$\frac{\sqrt[5]{{(x^{3})}^{4}}}{x^{3}}$

Multiply the exponents in ${(x^{3})}^{4}$ .

Tap for more steps...

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

$\frac{\sqrt[5]{x^{3 \cdot 4}}}{x^{3}}$

Multiply $3$ by $4$ .

$\frac{\sqrt[5]{x^{12}}}{x^{3}}$

Rewrite $x^{12}$ as ${(x^{2})}^{5} x^{2}$ .

Tap for more steps...

Factor out $x^{10}$ .

$\frac{\sqrt[5]{x^{10} x^{2}}}{x^{3}}$

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

$\frac{\sqrt[5]{{(x^{2})}^{5} x^{2}}}{x^{3}}$

Pull terms out from under the radical.

$\frac{x^{2} \sqrt[5]{x^{2}}}{x^{3}}$

Step 6

Cancel the common factor of $x^{2}$ and $x^{3}$ .

Tap for more steps...

Factor $x^{2}$ out of $x^{2} \sqrt[5]{x^{2}}$ .

$\frac{x^{2} (\sqrt[5]{x^{2}})}{x^{3}}$

Cancel the common factors.

Tap for more steps...

Factor $x^{2}$ out of $x^{3}$ .

$\frac{x^{2} (\sqrt[5]{x^{2}})}{x^{2} x}$

Cancel the common factor.

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

Rewrite the expression.

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