Simplify fourth root of 7x^6* fourth root of 32x^2

$\sqrt[4]{7 x^{6}} \cdot \sqrt[4]{32 x^{2}}$

Step 1

Rewrite $7 x^{6}$ as $x^{4} \cdot (7 x^{2})$ .

Factor out $x^{4}$ .

$\sqrt[4]{7 (x^{4} x^{2})} \cdot \sqrt[4]{32 x^{2}}$

Reorder $7$ and $x^{4}$ .

$\sqrt[4]{x^{4} \cdot 7 x^{2}} \cdot \sqrt[4]{32 x^{2}}$

Add parentheses.

$\sqrt[4]{x^{4} \cdot (7 x^{2})} \cdot \sqrt[4]{32 x^{2}}$

Step 2

Pull terms out from under the radical.

$x \sqrt[4]{7 x^{2}} \cdot \sqrt[4]{32 x^{2}}$

Step 3

Rewrite $32 x^{2}$ as $2^{4} \cdot (2 x^{2})$ .

Factor $16$ out of $32$ .

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

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

$x \sqrt[4]{7 x^{2}} \cdot \sqrt[4]{2^{4} \cdot 2 x^{2}}$

Add parentheses.

$x \sqrt[4]{7 x^{2}} \cdot \sqrt[4]{2^{4} \cdot (2 x^{2})}$

Step 4

Pull terms out from under the radical.

$x \sqrt[4]{7 x^{2}} \cdot (2 \sqrt[4]{2 x^{2}})$

Step 5

Rewrite using the commutative property of multiplication.

$2 x \sqrt[4]{7 x^{2}} \sqrt[4]{2 x^{2}}$

Step 6

Multiply $2 x \sqrt[4]{7 x^{2}} \sqrt[4]{2 x^{2}}$ .

Combine using the product rule for radicals.

$2 x \sqrt[4]{2 x^{2} (7 x^{2})}$

Multiply $7$ by $2$ .

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

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

Move $x^{2}$ .

$2 x \sqrt[4]{14 (x^{2} x^{2})}$

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

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

Add $2$ and $2$ .

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

Step 7

Reorder $14$ and $x^{4}$ .

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

Step 8

Pull terms out from under the radical.

$2 x (x \sqrt[4]{14})$

Step 9

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$2 (x \cdot x) \sqrt[4]{14}$

Multiply $x$ by $x$ .

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

Simplify fourth root of 7x^6* fourth root of 32x^2

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