Simplify (2p)/( square root of 16p^5)

$\frac{2 p}{\sqrt{16 p^{5}}}$

Step 1

Simplify the denominator.

Rewrite $16 p^{5}$ as ${(4 p^{2})}^{2} p$ .

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

$\frac{2 p}{\sqrt{4^{2} p^{5}}}$

Factor out $p^{4}$ .

$\frac{2 p}{\sqrt{4^{2} (p^{4} p)}}$

Rewrite $p^{4}$ as ${(p^{2})}^{2}$ .

$\frac{2 p}{\sqrt{4^{2} ({(p^{2})}^{2} p)}}$

Rewrite $4^{2} {(p^{2})}^{2}$ as ${(4 p^{2})}^{2}$ .

$\frac{2 p}{\sqrt{{(4 p^{2})}^{2} p}}$

Pull terms out from under the radical.

$\frac{2 p}{4 p^{2} \sqrt{p}}$

Step 2

Reduce the expression by cancelling the common factors.

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

Factor $2$ out of $2 p$ .

$\frac{2 (p)}{4 p^{2} \sqrt{p}}$

Cancel the common factors.

Factor $2$ out of $4 p^{2} \sqrt{p}$ .

$\frac{2 (p)}{2 (2 p^{2} \sqrt{p})}$

Cancel the common factor.

$\frac{2 p}{2 (2 p^{2} \sqrt{p})}$

Rewrite the expression.

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

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

Raise $p$ to the power of $1$ .

$\frac{p^{1}}{2 p^{2} \sqrt{p}}$

Factor $p$ out of $p^{1}$ .

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

Cancel the common factors.

Factor $p$ out of $2 p^{2} \sqrt{p}$ .

$\frac{p \cdot 1}{p (2 p \sqrt{p})}$

Cancel the common factor.

$\frac{p \cdot 1}{p (2 p \sqrt{p})}$

Rewrite the expression.

$\frac{1}{2 p \sqrt{p}}$

Step 3

Multiply $\frac{1}{2 p \sqrt{p}}$ by $\frac{\sqrt{p}}{\sqrt{p}}$ .

$\frac{1}{2 p \sqrt{p}} \cdot \frac{\sqrt{p}}{\sqrt{p}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{1}{2 p \sqrt{p}}$ by $\frac{\sqrt{p}}{\sqrt{p}}$ .

$\frac{\sqrt{p}}{2 p \sqrt{p} \sqrt{p}}$

Move $\sqrt{p}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 5

Multiply $p$ by $p$ by adding the exponents.

Move $p$ .

$\frac{\sqrt{p}}{2 (p \cdot p)}$

Multiply $p$ by $p$ .

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

Simplify (2p)/( square root of 16p^5)

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