Simplify fourth root of (3x^6y^5)/(16x^-2y^12)

$\sqrt[4]{\frac{3 x^{6} y^{5}}{16 x^{- 2} y^{12}}}$

Step 1

Reduce the expression $\frac{3 x^{6} y^{5}}{16 x^{- 2} y^{12}}$ by cancelling the common factors.

Factor $x^{- 2}$ out of $3 x^{6} y^{5}$ .

$\sqrt[4]{\frac{x^{- 2} (3 x^{8} y^{5})}{16 x^{- 2} y^{12}}}$

Factor $x^{- 2}$ out of $16 x^{- 2} y^{12}$ .

$\sqrt[4]{\frac{x^{- 2} (3 x^{8} y^{5})}{x^{- 2} (16 y^{12})}}$

Cancel the common factor.

$\sqrt[4]{\frac{x^{- 2} (3 x^{8} y^{5})}{x^{- 2} (16 y^{12})}}$

Rewrite the expression.

$\sqrt[4]{\frac{3 x^{8} y^{5}}{16 y^{12}}}$

Step 2

Cancel the common factor of $y^{5}$ and $y^{12}$ .

Factor $y^{5}$ out of $3 x^{8} y^{5}$ .

$\sqrt[4]{\frac{y^{5} (3 x^{8})}{16 y^{12}}}$

Cancel the common factors.

Factor $y^{5}$ out of $16 y^{12}$ .

$\sqrt[4]{\frac{y^{5} (3 x^{8})}{y^{5} (16 y^{7})}}$

Cancel the common factor.

$\sqrt[4]{\frac{y^{5} (3 x^{8})}{y^{5} (16 y^{7})}}$

Rewrite the expression.

$\sqrt[4]{\frac{3 x^{8}}{16 y^{7}}}$

Step 3

Rewrite $\frac{3 x^{8}}{16 y^{7}}$ as ${(\frac{x^{2}}{2 y})}^{4} \frac{3}{y^{3}}$ .

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

$\sqrt[4]{\frac{{(x^{2})}^{4} \cdot 3}{16 y^{7}}}$

Factor the perfect power ${(2 y)}^{4}$ out of $16 y^{7}$ .

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

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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

Step 6

Combine.

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

Step 7

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

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

Step 8

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $3$ .

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

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 $y^{\frac{3}{4}}$ .

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

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

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

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

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

Multiply $3$ by $4$ .

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

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

Factor $4$ out of $12$ .

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

Cancel the common factors.

Factor $4$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $3$ by $1$ .

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

Step 9

Multiply $y$ by $y^{3}$ by adding the exponents.

Move $y^{3}$ .

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

Multiply $y^{3}$ by $y$ .

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

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

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

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

Add $3$ and $1$ .

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

Step 10

Simplify the numerator.

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

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

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

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

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

Multiply $3$ by $3$ .

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

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

Factor out $y^{8}$ .

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

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

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

Pull terms out from under the radical.

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

Combine using the product rule for radicals.

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

Step 11

Cancel the common factor of $y^{2}$ and $y^{4}$ .

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

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

Cancel the common factors.

Factor $y^{2}$ out of $2 y^{4}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify fourth root of (3x^6y^5)/(16x^-2y^12)

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