Simplify (2x^-2y^2)/( fourth root of 81x^4y^3)

$\frac{2 x^{- 2} y^{2}}{\sqrt[4]{81 x^{4} y^{3}}}$

Step 1

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Simplify the denominator.

Rewrite $81 x^{4} y^{3}$ as ${(3 x)}^{4} y^{3}$ .

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

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

Rewrite $3^{4} x^{4}$ as ${(3 x)}^{4}$ .

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

Pull terms out from under the radical.

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

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

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

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

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

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

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

Add $2$ and $1$ .

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

Step 3

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

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

Step 4

Simplify terms.

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $3$ .

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

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

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

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

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

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

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

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

Multiply $3$ by $4$ .

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

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

Factor $4$ out of $12$ .

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

Cancel the common factors.

Factor $4$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $3$ by $1$ .

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

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

Factor $y^{2}$ out of $2 y^{2} {\sqrt[4]{y^{3}}}^{3}$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Simplify the numerator.

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

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

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

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

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

Multiply $3$ by $3$ .

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

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

Factor out $y^{8}$ .

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

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

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

Pull terms out from under the radical.

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

Step 6

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (2x^-2y^2)/( fourth root of 81x^4y^3)

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