Simplify (4v^2)/( square root of 8v^6)

$\frac{4 v^{2}}{\sqrt{8 v^{6}}}$

Step 1

Simplify the denominator.

Rewrite $8 v^{6}$ as ${(2 v^{3})}^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$\frac{4 v^{2}}{\sqrt{4 (2) v^{6}}}$

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

$\frac{4 v^{2}}{\sqrt{2^{2} \cdot 2 v^{6}}}$

Rewrite $v^{6}$ as ${(v^{3})}^{2}$ .

$\frac{4 v^{2}}{\sqrt{2^{2} \cdot 2 {(v^{3})}^{2}}}$

Move $2$ .

$\frac{4 v^{2}}{\sqrt{2^{2} {(v^{3})}^{2} \cdot 2}}$

Rewrite $2^{2} {(v^{3})}^{2}$ as ${(2 v^{3})}^{2}$ .

$\frac{4 v^{2}}{\sqrt{{(2 v^{3})}^{2} \cdot 2}}$

Pull terms out from under the radical.

$\frac{4 v^{2}}{2 v^{3} \sqrt{2}}$

Step 2

Reduce the expression by cancelling the common factors.

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4 v^{2}$ .

$\frac{2 (2 v^{2})}{2 v^{3} \sqrt{2}}$

Cancel the common factors.

Factor $2$ out of $2 v^{3} \sqrt{2}$ .

$\frac{2 (2 v^{2})}{2 (v^{3} \sqrt{2})}$

Cancel the common factor.

$\frac{2 (2 v^{2})}{2 (v^{3} \sqrt{2})}$

Rewrite the expression.

$\frac{2 v^{2}}{v^{3} \sqrt{2}}$

Cancel the common factor of $v^{2}$ and $v^{3}$ .

Factor $v^{2}$ out of $2 v^{2}$ .

$\frac{v^{2} \cdot 2}{v^{3} \sqrt{2}}$

Cancel the common factors.

Factor $v^{2}$ out of $v^{3} \sqrt{2}$ .

$\frac{v^{2} \cdot 2}{v^{2} (v \sqrt{2})}$

Cancel the common factor.

$\frac{v^{2} \cdot 2}{v^{2} (v \sqrt{2})}$

Rewrite the expression.

$\frac{2}{v \sqrt{2}}$

Step 3

Multiply $\frac{2}{v \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{2}{v \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{2}{v \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{2 \sqrt{2}}{v \sqrt{2} \sqrt{2}}$

Move $\sqrt{2}$ .

$\frac{2 \sqrt{2}}{v (\sqrt{2} \sqrt{2})}$

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

$\frac{2 \sqrt{2}}{v ({\sqrt{2}}^{1} \sqrt{2})}$

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

$\frac{2 \sqrt{2}}{v ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

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

$\frac{2 \sqrt{2}}{v {\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{2 \sqrt{2}}{v {\sqrt{2}}^{2}}$

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

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

$\frac{2 \sqrt{2}}{v {(2^{\frac{1}{2}})}^{2}}$

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

$\frac{2 \sqrt{2}}{v \cdot 2^{\frac{1}{2} \cdot 2}}$

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

$\frac{2 \sqrt{2}}{v \cdot 2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 \sqrt{2}}{v \cdot 2^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{2 \sqrt{2}}{v \cdot 2^{1}}$

Evaluate the exponent.

$\frac{2 \sqrt{2}}{v \cdot 2}$

Step 5

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 \sqrt{2}}{v \cdot 2}$

Rewrite the expression.

$\frac{\sqrt{2}}{v}$

Simplify (4v^2)/( square root of 8v^6)

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