Simplify ( eighth root of 16x^12)^6

${(\sqrt[8]{16 x^{12}})}^{6}$

Step 1

Rewrite $16 x^{12}$ as $x^{8} \cdot (16 x^{4})$ .

Factor out $x^{8}$ .

${\sqrt[8]{16 (x^{8} x^{4})}}^{6}$

Reorder $16$ and $x^{8}$ .

${\sqrt[8]{x^{8} \cdot 16 x^{4}}}^{6}$

Add parentheses.

${\sqrt[8]{x^{8} \cdot (16 x^{4})}}^{6}$

Step 2

Pull terms out from under the radical.

${(x \sqrt[8]{16 x^{4}})}^{6}$

Step 3

Rewrite $16 x^{4}$ as ${(2 x)}^{4}$ .

${(x \sqrt[8]{{(2 x)}^{4}})}^{6}$

Step 4

Rewrite $\sqrt[8]{{(2 x)}^{4}}$ as $\sqrt{\sqrt[4]{{(2 x)}^{4}}}$ .

${(x \sqrt{\sqrt[4]{{(2 x)}^{4}}})}^{6}$

Step 5

Pull terms out from under the radical, assuming positive real numbers.

${(x \sqrt{2 x})}^{6}$

Step 6

Apply the product rule to $x \sqrt{2 x}$ .

$x^{6} {\sqrt{2 x}}^{6}$

Step 7

Rewrite ${\sqrt{2 x}}^{6}$ as ${(2 x)}^{3}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x}$ as ${(2 x)}^{\frac{1}{2}}$ .

$x^{6} {({(2 x)}^{\frac{1}{2}})}^{6}$

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

$x^{6} {(2 x)}^{\frac{1}{2} \cdot 6}$

Combine $\frac{1}{2}$ and $6$ .

$x^{6} {(2 x)}^{\frac{6}{2}}$

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $6$ .

$x^{6} {(2 x)}^{\frac{2 \cdot 3}{2}}$

Cancel the common factors.

Factor $2$ out of $2$ .

$x^{6} {(2 x)}^{\frac{2 \cdot 3}{2 (1)}}$

Cancel the common factor.

$x^{6} {(2 x)}^{\frac{2 \cdot 3}{2 \cdot 1}}$

Rewrite the expression.

$x^{6} {(2 x)}^{\frac{3}{1}}$

Divide $3$ by $1$ .

$x^{6} {(2 x)}^{3}$

Step 8

Simplify the expression.

Apply the product rule to $2 x$ .

$x^{6} (2^{3} x^{3})$

Rewrite using the commutative property of multiplication.

$2^{3} x^{6} x^{3}$

Step 9

Multiply $x^{6}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$2^{3} (x^{3} x^{6})$

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

$2^{3} x^{3 + 6}$

Add $3$ and $6$ .

$2^{3} x^{9}$

Step 10

Raise $2$ to the power of $3$ .

$8 x^{9}$

Simplify ( eighth root of 16x^12)^6

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