Algebra Examples

$\sqrt[4]{\frac{5}{x^{3} y^{8}}}$

Step 1

Rewrite $\frac{5}{x^{3} y^{8}}$ as ${(\frac{1}{y^{2}})}^{4} \frac{5}{x^{3}}$ .

Tap for more steps...

Step 1.1

Factor the perfect power $1^{4}$ out of $5$ .

$\sqrt[4]{\frac{1^{4} \cdot 5}{x^{3} y^{8}}}$

Step 1.2

Factor the perfect power ${(y^{2})}^{4}$ out of $x^{3} y^{8}$ .

$\sqrt[4]{\frac{1^{4} \cdot 5}{{(y^{2})}^{4} x^{3}}}$

Step 1.3

Rearrange the fraction $\frac{1^{4} \cdot 5}{{(y^{2})}^{4} x^{3}}$ .

$\sqrt[4]{{(\frac{1}{y^{2}})}^{4} \frac{5}{x^{3}}}$

Step 2

Pull terms out from under the radical.

$| \frac{1}{y^{2}} | \sqrt[4]{\frac{5}{x^{3}}}$

Step 3

Remove non-negative terms from the absolute value.

$\frac{| 1 |}{y^{2}} \sqrt[4]{\frac{5}{x^{3}}}$

Step 4

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{1}{y^{2}} \sqrt[4]{\frac{5}{x^{3}}}$

Step 5

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

$\frac{1}{y^{2}} \cdot \frac{\sqrt[4]{5}}{\sqrt[4]{x^{3}}}$

Step 6

Combine.

$\frac{1 \sqrt[4]{5}}{y^{2} \sqrt[4]{x^{3}}}$

Step 7

Multiply $\sqrt[4]{5}$ by $1$ .

$\frac{\sqrt[4]{5}}{y^{2} \sqrt[4]{x^{3}}}$

Step 8

Multiply $\frac{\sqrt[4]{5}}{y^{2} \sqrt[4]{x^{3}}}$ by $\frac{{\sqrt[4]{x^{3}}}^{3}}{{\sqrt[4]{x^{3}}}^{3}}$ .

$\frac{\sqrt[4]{5}}{y^{2} \sqrt[4]{x^{3}}} \cdot \frac{{\sqrt[4]{x^{3}}}^{3}}{{\sqrt[4]{x^{3}}}^{3}}$

Step 9

Combine and simplify the denominator.

Tap for more steps...

Step 9.1

Multiply $\frac{\sqrt[4]{5}}{y^{2} \sqrt[4]{x^{3}}}$ by $\frac{{\sqrt[4]{x^{3}}}^{3}}{{\sqrt[4]{x^{3}}}^{3}}$ .

$\frac{\sqrt[4]{5} {\sqrt[4]{x^{3}}}^{3}}{y^{2} \sqrt[4]{x^{3}} {\sqrt[4]{x^{3}}}^{3}}$

Step 9.2

Move $\sqrt[4]{x^{3}}$ .

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

Step 9.3

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

$\frac{\sqrt[4]{5} {\sqrt[4]{x^{3}}}^{3}}{y^{2} ({\sqrt[4]{x^{3}}}^{1} {\sqrt[4]{x^{3}}}^{3})}$

Step 9.4

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

$\frac{\sqrt[4]{5} {\sqrt[4]{x^{3}}}^{3}}{y^{2} {\sqrt[4]{x^{3}}}^{1 + 3}}$

Step 9.5

Add $1$ and $3$ .

$\frac{\sqrt[4]{5} {\sqrt[4]{x^{3}}}^{3}}{y^{2} {\sqrt[4]{x^{3}}}^{4}}$

Step 9.6

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

Tap for more steps...

Step 9.6.1

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

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

Step 9.6.2

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

$\frac{\sqrt[4]{5} {\sqrt[4]{x^{3}}}^{3}}{y^{2} x^{\frac{3}{4} \cdot 4}}$

Step 9.6.3

Combine $\frac{3}{4}$ and $4$ .

$\frac{\sqrt[4]{5} {\sqrt[4]{x^{3}}}^{3}}{y^{2} x^{\frac{3 \cdot 4}{4}}}$

Step 9.6.4

Multiply $3$ by $4$ .

$\frac{\sqrt[4]{5} {\sqrt[4]{x^{3}}}^{3}}{y^{2} x^{\frac{12}{4}}}$

Step 9.6.5

Cancel the common factor of $12$ and $4$ .

Tap for more steps...

Step 9.6.5.1

Factor $4$ out of $12$ .

$\frac{\sqrt[4]{5} {\sqrt[4]{x^{3}}}^{3}}{y^{2} x^{\frac{4 \cdot 3}{4}}}$

Step 9.6.5.2

Cancel the common factors.

Tap for more steps...

Step 9.6.5.2.1

Factor $4$ out of $4$ .

$\frac{\sqrt[4]{5} {\sqrt[4]{x^{3}}}^{3}}{y^{2} x^{\frac{4 \cdot 3}{4 (1)}}}$

Step 9.6.5.2.2

Cancel the common factor.

$\frac{\sqrt[4]{5} {\sqrt[4]{x^{3}}}^{3}}{y^{2} x^{\frac{4 \cdot 3}{4 \cdot 1}}}$

Step 9.6.5.2.3

Rewrite the expression.

$\frac{\sqrt[4]{5} {\sqrt[4]{x^{3}}}^{3}}{y^{2} x^{\frac{3}{1}}}$

Step 9.6.5.2.4

Divide $3$ by $1$ .

$\frac{\sqrt[4]{5} {\sqrt[4]{x^{3}}}^{3}}{y^{2} x^{3}}$

Step 10

Simplify the numerator.

Tap for more steps...

Step 10.1

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

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

Step 10.2

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

Tap for more steps...

Step 10.2.1

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

$\frac{\sqrt[4]{5} \sqrt[4]{x^{3 \cdot 3}}}{y^{2} x^{3}}$

Step 10.2.2

Multiply $3$ by $3$ .

$\frac{\sqrt[4]{5} \sqrt[4]{x^{9}}}{y^{2} x^{3}}$

Step 10.3

Rewrite $x^{9}$ as ${(x^{2})}^{4} x$ .

Tap for more steps...

Step 10.3.1

Factor out $x^{8}$ .

$\frac{\sqrt[4]{5} \sqrt[4]{x^{8} x}}{y^{2} x^{3}}$

Step 10.3.2

Rewrite $x^{8}$ as ${(x^{2})}^{4}$ .

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

Step 10.4

Pull terms out from under the radical.

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

Step 10.5

Remove non-negative terms from the absolute value.

$\frac{\sqrt[4]{5} x^{2} \sqrt[4]{x}}{y^{2} x^{3}}$

Step 10.6

Combine using the product rule for radicals.

$\frac{x^{2} \sqrt[4]{5 x}}{y^{2} x^{3}}$

Step 11

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

Tap for more steps...

Step 11.1

Factor $x^{2}$ out of $x^{2} \sqrt[4]{5 x}$ .

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

Step 11.2

Cancel the common factors.

Tap for more steps...

Step 11.2.1

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

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

Step 11.2.2

Cancel the common factor.

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

Step 11.2.3

Rewrite the expression.

$\frac{\sqrt[4]{5 x}}{y^{2} x}$