Simplify the Radical Expression ( fourth root of 16x^16y^24)/( fourth root of x^2y^4)

$\frac{\sqrt[4]{16 x^{16} y^{24}}}{\sqrt[4]{x^{2} y^{4}}}$

Step 1

Combine $\sqrt[4]{16 x^{16} y^{24}}$ and $\sqrt[4]{x^{2} y^{4}}$ into a single radical.

$\sqrt[4]{\frac{16 x^{16} y^{24}}{x^{2} y^{4}}}$

Step 2

Reduce the expression $\frac{16 x^{16} y^{24}}{x^{2} y^{4}}$ by cancelling the common factors.

Factor $x^{2}$ out of $16 x^{16} y^{24}$ .

$\sqrt[4]{\frac{x^{2} (16 x^{14} y^{24})}{x^{2} y^{4}}}$

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

$\sqrt[4]{\frac{x^{2} (16 x^{14} y^{24})}{x^{2} (y^{4})}}$

Cancel the common factor.

$\sqrt[4]{\frac{x^{2} (16 x^{14} y^{24})}{x^{2} y^{4}}}$

Rewrite the expression.

$\sqrt[4]{\frac{16 x^{14} y^{24}}{y^{4}}}$

Step 3

Cancel the common factor of $y^{24}$ and $y^{4}$ .

Factor $y^{4}$ out of $16 x^{14} y^{24}$ .

$\sqrt[4]{\frac{y^{4} (16 x^{14} y^{20})}{y^{4}}}$

Cancel the common factors.

Multiply by $1$ .

$\sqrt[4]{\frac{y^{4} (16 x^{14} y^{20})}{y^{4} \cdot 1}}$

Cancel the common factor.

$\sqrt[4]{\frac{y^{4} (16 x^{14} y^{20})}{y^{4} \cdot 1}}$

Rewrite the expression.

$\sqrt[4]{\frac{16 x^{14} y^{20}}{1}}$

Divide $16 x^{14} y^{20}$ by $1$ .

$\sqrt[4]{16 x^{14} y^{20}}$

Step 4

Rewrite $16 x^{14} y^{20}$ as ${(2 x^{3} y^{5})}^{4} x^{2}$ .

Rewrite $16$ as $2^{4}$ .

$\sqrt[4]{2^{4} x^{14} y^{20}}$

Factor out $x^{12}$ .

$\sqrt[4]{2^{4} (x^{12} x^{2}) y^{20}}$

Rewrite $x^{12}$ as ${(x^{3})}^{4}$ .

$\sqrt[4]{2^{4} ({(x^{3})}^{4} x^{2}) y^{20}}$

Rewrite $y^{20}$ as ${(y^{5})}^{4}$ .

$\sqrt[4]{2^{4} ({(x^{3})}^{4} x^{2}) {(y^{5})}^{4}}$

Move $x^{2}$ .

$\sqrt[4]{2^{4} ({(x^{3})}^{4}) {(y^{5})}^{4} x^{2}}$

Rewrite $2^{4} ({(x^{3})}^{4}) {(y^{5})}^{4}$ as ${(2 x^{3} y^{5})}^{4}$ .

$\sqrt[4]{{(2 x^{3} y^{5})}^{4} x^{2}}$

Step 5

Pull terms out from under the radical.

$| 2 x^{3} y^{5} | \sqrt[4]{x^{2}}$

Step 6

Remove non-negative terms from the absolute value.

$2 | x^{3} y^{5} | \sqrt[4]{x^{2}}$

Step 7

Rewrite $\sqrt[4]{x^{2}}$ as $\sqrt{\sqrt{x^{2}}}$ .

$2 | x^{3} y^{5} | \sqrt{\sqrt{x^{2}}}$

Step 8

Pull terms out from under the radical.

$2 | x^{3} y^{5} | \sqrt{| x |}$

Simplify the Radical Expression ( fourth root of 16x^16y^24)/( fourth root of x^2y^4)

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