Basic Math Examples

$\frac{\sqrt[4]{256}}{\sqrt[4]{9 y^{5}}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $256$ as $4^{4}$ .

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

Step 1.2

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

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

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $9 y^{5}$ as $y^{4} \cdot (9 y)$ .

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Step 2.1.1

Factor out $y^{4}$ .

$\frac{4}{\sqrt[4]{9 (y^{4} y)}}$

Step 2.1.2

Reorder $9$ and $y^{4}$ .

$\frac{4}{\sqrt[4]{y^{4} \cdot 9 y}}$

Step 2.1.3

Add parentheses.

$\frac{4}{\sqrt[4]{y^{4} \cdot (9 y)}}$

Step 2.2

Pull terms out from under the radical.

$\frac{4}{y \sqrt[4]{9 y}}$

Step 3

Multiply $\frac{4}{y \sqrt[4]{9 y}}$ by $\frac{{\sqrt[4]{9 y}}^{3}}{{\sqrt[4]{9 y}}^{3}}$ .

$\frac{4}{y \sqrt[4]{9 y}} \cdot \frac{{\sqrt[4]{9 y}}^{3}}{{\sqrt[4]{9 y}}^{3}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{4}{y \sqrt[4]{9 y}}$ by $\frac{{\sqrt[4]{9 y}}^{3}}{{\sqrt[4]{9 y}}^{3}}$ .

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

Step 4.2

Move $\sqrt[4]{9 y}$ .

$\frac{4 {\sqrt[4]{9 y}}^{3}}{y (\sqrt[4]{9 y} {\sqrt[4]{9 y}}^{3})}$

Step 4.3

Raise $\sqrt[4]{9 y}$ to the power of $1$ .

$\frac{4 {\sqrt[4]{9 y}}^{3}}{y ({\sqrt[4]{9 y}}^{1} {\sqrt[4]{9 y}}^{3})}$

Step 4.4

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

$\frac{4 {\sqrt[4]{9 y}}^{3}}{y {\sqrt[4]{9 y}}^{1 + 3}}$

Step 4.5

Add $1$ and $3$ .

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

Step 4.6

Rewrite ${\sqrt[4]{9 y}}^{4}$ as $9 y$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{9 y}$ as ${(9 y)}^{\frac{1}{4}}$ .

$\frac{4 {\sqrt[4]{9 y}}^{3}}{y {({(9 y)}^{\frac{1}{4}})}^{4}}$

Step 4.6.2

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

$\frac{4 {\sqrt[4]{9 y}}^{3}}{y {(9 y)}^{\frac{1}{4} \cdot 4}}$

Step 4.6.3

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

$\frac{4 {\sqrt[4]{9 y}}^{3}}{y {(9 y)}^{\frac{4}{4}}}$

Step 4.6.4

Cancel the common factor of $4$ .

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Step 4.6.4.1

Cancel the common factor.

$\frac{4 {\sqrt[4]{9 y}}^{3}}{y {(9 y)}^{\frac{4}{4}}}$

Step 4.6.4.2

Rewrite the expression.

$\frac{4 {\sqrt[4]{9 y}}^{3}}{y {(9 y)}^{1}}$

Step 4.6.5

Simplify.

$\frac{4 {\sqrt[4]{9 y}}^{3}}{y (9 y)}$

Step 5

Multiply $y$ by $y$ by adding the exponents.

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Step 5.1

Move $y$ .

$\frac{4 {\sqrt[4]{9 y}}^{3}}{y \cdot y \cdot 9}$

Step 5.2

Multiply $y$ by $y$ .

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

Step 6

Simplify the numerator.

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Step 6.1

Rewrite ${\sqrt[4]{9 y}}^{3}$ as $\sqrt[4]{{(9 y)}^{3}}$ .

$\frac{4 \sqrt[4]{{(9 y)}^{3}}}{y^{2} \cdot 9}$

Step 6.2

Apply the product rule to $9 y$ .

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

Step 6.3

Raise $9$ to the power of $3$ .

$\frac{4 \sqrt[4]{729 y^{3}}}{y^{2} \cdot 9}$

Step 6.4

Rewrite $729 y^{3}$ as $3^{4} \cdot (9 y^{3})$ .

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Step 6.4.1

Factor $81$ out of $729$ .

$\frac{4 \sqrt[4]{81 (9) y^{3}}}{y^{2} \cdot 9}$

Step 6.4.2

Rewrite $81$ as $3^{4}$ .

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

Step 6.4.3

Add parentheses.

$\frac{4 \sqrt[4]{3^{4} \cdot (9 y^{3})}}{y^{2} \cdot 9}$

Step 6.5

Pull terms out from under the radical.

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

Step 6.6

Multiply $4$ by $3$ .

$\frac{12 \sqrt[4]{9 y^{3}}}{y^{2} \cdot 9}$

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Cancel the common factor of $12$ and $9$ .

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Step 7.1.1

Factor $3$ out of $12 \sqrt[4]{9 y^{3}}$ .

$\frac{3 (4 \sqrt[4]{9 y^{3}})}{y^{2} \cdot 9}$

Step 7.1.2

Cancel the common factors.

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Step 7.1.2.1

Factor $3$ out of $y^{2} \cdot 9$ .

$\frac{3 (4 \sqrt[4]{9 y^{3}})}{3 (y^{2} \cdot 3)}$

Step 7.1.2.2

Cancel the common factor.

$\frac{3 (4 \sqrt[4]{9 y^{3}})}{3 (y^{2} \cdot 3)}$

Step 7.1.2.3

Rewrite the expression.

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

Step 7.2

Move $3$ to the left of $y^{2}$ .

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