Simplify the Radical Expression fourth root of 64x^4y^16

$\sqrt[4]{64 x^{4} y^{16}}$

Step 1

Rewrite $64 x^{4} y^{16}$ as ${(2 x y^{4})}^{4} \cdot 4$ .

Factor $16$ out of $64$ .

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

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

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

Rewrite $y^{16}$ as ${(y^{4})}^{4}$ .

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

Move $4$ .

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

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

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

Step 2

Pull terms out from under the radical.

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

Step 3

Remove non-negative terms from the absolute value.

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

Step 4

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

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

Step 5

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

$2 y^{4} | x | \sqrt{\sqrt{2^{2}}}$

Step 6

Pull terms out from under the radical.

$2 y^{4} | x | \sqrt{| 2 |}$

Step 7

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$2 y^{4} | x | \sqrt{2}$