Simplify 4/( cube root of 9y^4)

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

Step 1

Simplify the denominator.

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

Factor out $y^{3}$ .

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

Reorder $9$ and $y^{3}$ .

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

Add parentheses.

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

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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

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

Move $\sqrt[3]{9 y}$ .

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

Raise $\sqrt[3]{9 y}$ to the power of $1$ .

$\frac{4 {\sqrt[3]{9 y}}^{2}}{y ({\sqrt[3]{9 y}}^{1} {\sqrt[3]{9 y}}^{2})}$

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

$\frac{4 {\sqrt[3]{9 y}}^{2}}{y {\sqrt[3]{9 y}}^{1 + 2}}$

Add $1$ and $2$ .

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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9 y}$ as ${(9 y)}^{\frac{1}{3}}$ .

$\frac{4 {\sqrt[3]{9 y}}^{2}}{y {({(9 y)}^{\frac{1}{3}})}^{3}}$

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

$\frac{4 {\sqrt[3]{9 y}}^{2}}{y {(9 y)}^{\frac{1}{3} \cdot 3}}$

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{4 {\sqrt[3]{9 y}}^{2}}{y {(9 y)}^{1}}$

Simplify.

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

Step 4

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

Move $y$ .

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

Multiply $y$ by $y$ .

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

Step 5

Simplify the numerator.

Rewrite ${\sqrt[3]{9 y}}^{2}$ as $\sqrt[3]{{(9 y)}^{2}}$ .

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

Apply the product rule to $9 y$ .

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

Raise $9$ to the power of $2$ .

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

Rewrite $81 y^{2}$ as $3^{3} \cdot (3 y^{2})$ .

Factor $27$ out of $81$ .

$\frac{4 \sqrt[3]{27 (3) y^{2}}}{y^{2} \cdot 9}$

Rewrite $27$ as $3^{3}$ .

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

Add parentheses.

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

Pull terms out from under the radical.

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

Multiply $4$ by $3$ .

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

Step 6

Reduce the expression by cancelling the common factors.

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

Factor $3$ out of $12 \sqrt[3]{3 y^{2}}$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Simplify 4/( cube root of 9y^4)

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