Simplify square root of 1/128c^4d^7

$\sqrt{\frac{1}{128} c^{4} d^{7}}$

Step 1

Combine $\frac{1}{128}$ and $c^{4}$ .

$\sqrt{\frac{c^{4}}{128} d^{7}}$

Step 2

Combine $\frac{c^{4}}{128}$ and $d^{7}$ .

$\sqrt{\frac{c^{4} d^{7}}{128}}$

Step 3

Rewrite $\frac{c^{4} d^{7}}{128}$ as ${(\frac{c^{2} d^{3}}{8})}^{2} \frac{d}{2}$ .

Factor the perfect power ${(c^{2} d^{3})}^{2}$ out of $c^{4} d^{7}$ .

$\sqrt{\frac{{(c^{2} d^{3})}^{2} d}{128}}$

Factor the perfect power $8^{2}$ out of $128$ .

$\sqrt{\frac{{(c^{2} d^{3})}^{2} d}{8^{2} \cdot 2}}$

Rearrange the fraction $\frac{{(c^{2} d^{3})}^{2} d}{8^{2} \cdot 2}$ .

$\sqrt{{(\frac{c^{2} d^{3}}{8})}^{2} \frac{d}{2}}$

Step 4

Pull terms out from under the radical.

$\frac{c^{2} d^{3}}{8} \sqrt{\frac{d}{2}}$

Step 5

Rewrite $\sqrt{\frac{d}{2}}$ as $\frac{\sqrt{d}}{\sqrt{2}}$ .

$\frac{c^{2} d^{3}}{8} \cdot \frac{\sqrt{d}}{\sqrt{2}}$

Step 6

Combine.

$\frac{c^{2} d^{3} \sqrt{d}}{8 \sqrt{2}}$

Step 7

Multiply $\frac{c^{2} d^{3} \sqrt{d}}{8 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{c^{2} d^{3} \sqrt{d}}{8 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 8

Combine and simplify the denominator.

Multiply $\frac{c^{2} d^{3} \sqrt{d}}{8 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{c^{2} d^{3} \sqrt{d} \sqrt{2}}{8 \sqrt{2} \sqrt{2}}$

Move $\sqrt{2}$ .

$\frac{c^{2} d^{3} \sqrt{d} \sqrt{2}}{8 (\sqrt{2} \sqrt{2})}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{c^{2} d^{3} \sqrt{d} \sqrt{2}}{8 ({\sqrt{2}}^{1} \sqrt{2})}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{c^{2} d^{3} \sqrt{d} \sqrt{2}}{8 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

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

$\frac{c^{2} d^{3} \sqrt{d} \sqrt{2}}{8 {\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{c^{2} d^{3} \sqrt{d} \sqrt{2}}{8 {\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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

$\frac{c^{2} d^{3} \sqrt{d} \sqrt{2}}{8 {(2^{\frac{1}{2}})}^{2}}$

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

$\frac{c^{2} d^{3} \sqrt{d} \sqrt{2}}{8 \cdot 2^{\frac{1}{2} \cdot 2}}$

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

$\frac{c^{2} d^{3} \sqrt{d} \sqrt{2}}{8 \cdot 2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{c^{2} d^{3} \sqrt{d} \sqrt{2}}{8 \cdot 2^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{c^{2} d^{3} \sqrt{d} \sqrt{2}}{8 \cdot 2^{1}}$

Evaluate the exponent.

$\frac{c^{2} d^{3} \sqrt{d} \sqrt{2}}{8 \cdot 2}$

Step 9

Combine using the product rule for radicals.

$\frac{c^{2} d^{3} \sqrt{2 d}}{8 \cdot 2}$

Step 10

Multiply $8$ by $2$ .

$\frac{c^{2} d^{3} \sqrt{2 d}}{16}$

Simplify square root of 1/128c^4d^7

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