Simplify cube root of (a^3)/(-512y)

$\sqrt[3]{\frac{a^{3}}{- 512 y}}$

Step 1

Move the negative in front of the fraction.

$\sqrt[3]{- \frac{a^{3}}{(512) y}}$

Step 2

Rewrite $- \frac{a^{3}}{512 y}$ as ${(- \frac{a}{2^{3}})}^{3} \frac{1}{y}$ .

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$\sqrt[3]{{(- 1)}^{3} \frac{a^{3}}{512 y}}$

Factor the perfect power $a^{3}$ out of $a^{3}$ .

$\sqrt[3]{{(- 1)}^{3} \frac{a^{3} \cdot 1}{512 y}}$

Factor the perfect power ${(2^{3})}^{3}$ out of $512 y$ .

$\sqrt[3]{{(- 1)}^{3} \frac{a^{3} \cdot 1}{{(2^{3})}^{3} y}}$

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

$\sqrt[3]{{(- 1)}^{3} ({(\frac{a}{2^{3}})}^{3} \frac{1}{y})}$

Rewrite ${(- 1)}^{3} {(\frac{a}{2^{3}})}^{3}$ as ${(- \frac{a}{2^{3}})}^{3}$ .

$\sqrt[3]{{(- \frac{a}{2^{3}})}^{3} \frac{1}{y}}$

Step 3

Pull terms out from under the radical.

$- \frac{a}{2^{3}} \sqrt[3]{\frac{1}{y}}$

Step 4

Raise $2$ to the power of $3$ .

$- \frac{a}{8} \sqrt[3]{\frac{1}{y}}$

Step 5

Rewrite $\sqrt[3]{\frac{1}{y}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{y}}$ .

$- \frac{a}{8} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{y}}$

Step 6

Any root of $1$ is $1$ .

$- \frac{a}{8} \cdot \frac{1}{\sqrt[3]{y}}$

Step 7

Multiply $\frac{1}{\sqrt[3]{y}}$ by $\frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}}$ .

$- \frac{a}{8} (\frac{1}{\sqrt[3]{y}} \cdot \frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}})$

Step 8

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt[3]{y}}$ by $\frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}}$ .

$- \frac{a}{8} \cdot \frac{{\sqrt[3]{y}}^{2}}{\sqrt[3]{y} {\sqrt[3]{y}}^{2}}$

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

$- \frac{a}{8} \cdot \frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{1} {\sqrt[3]{y}}^{2}}$

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

$- \frac{a}{8} \cdot \frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{1 + 2}}$

Add $1$ and $2$ .

$- \frac{a}{8} \cdot \frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{3}}$

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

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

$- \frac{a}{8} \cdot \frac{{\sqrt[3]{y}}^{2}}{{(y^{\frac{1}{3}})}^{3}}$

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

$- \frac{a}{8} \cdot \frac{{\sqrt[3]{y}}^{2}}{y^{\frac{1}{3} \cdot 3}}$

Combine $\frac{1}{3}$ and $3$ .

$- \frac{a}{8} \cdot \frac{{\sqrt[3]{y}}^{2}}{y^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$- \frac{a}{8} \cdot \frac{{\sqrt[3]{y}}^{2}}{y^{\frac{3}{3}}}$

Rewrite the expression.

$- \frac{a}{8} \cdot \frac{{\sqrt[3]{y}}^{2}}{y^{1}}$

Simplify.

$- \frac{a}{8} \cdot \frac{{\sqrt[3]{y}}^{2}}{y}$

Step 9

Rewrite ${\sqrt[3]{y}}^{2}$ as ${(y^{2})}^{\frac{1}{3}}$ .

$- \frac{a}{8} \cdot \frac{\sqrt[3]{y^{2}}}{y}$

Step 10

Multiply $\frac{\sqrt[3]{y^{2}}}{y}$ by $\frac{a}{8}$ .

$- \frac{\sqrt[3]{y^{2}} a}{y \cdot 8}$

Step 11

Move $8$ to the left of $y$ .

$- \frac{\sqrt[3]{y^{2}} a}{8 y}$

Simplify cube root of (a^3)/(-512y)

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