Simplify (2a)/( square root of 36a^7)

$\frac{2 a}{\sqrt{36 a^{7}}}$

Step 1

Simplify the denominator.

Rewrite $36 a^{7}$ as ${(6 a^{3})}^{2} a$ .

Rewrite $36$ as $6^{2}$ .

$\frac{2 a}{\sqrt{6^{2} a^{7}}}$

Factor out $a^{6}$ .

$\frac{2 a}{\sqrt{6^{2} (a^{6} a)}}$

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

$\frac{2 a}{\sqrt{6^{2} ({(a^{3})}^{2} a)}}$

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

$\frac{2 a}{\sqrt{{(6 a^{3})}^{2} a}}$

Pull terms out from under the radical.

$\frac{2 a}{6 a^{3} \sqrt{a}}$

Step 2

Reduce the expression by cancelling the common factors.

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

Factor $2$ out of $2 a$ .

$\frac{2 (a)}{6 a^{3} \sqrt{a}}$

Cancel the common factors.

Factor $2$ out of $6 a^{3} \sqrt{a}$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{a}{3 a^{3} \sqrt{a}}$

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

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

$\frac{a^{1}}{3 a^{3} \sqrt{a}}$

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

$\frac{a \cdot 1}{3 a^{3} \sqrt{a}}$

Cancel the common factors.

Factor $a$ out of $3 a^{3} \sqrt{a}$ .

$\frac{a \cdot 1}{a (3 a^{2} \sqrt{a})}$

Cancel the common factor.

$\frac{a \cdot 1}{a (3 a^{2} \sqrt{a})}$

Rewrite the expression.

$\frac{1}{3 a^{2} \sqrt{a}}$

Step 3

Multiply $\frac{1}{3 a^{2} \sqrt{a}}$ by $\frac{\sqrt{a}}{\sqrt{a}}$ .

$\frac{1}{3 a^{2} \sqrt{a}} \cdot \frac{\sqrt{a}}{\sqrt{a}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{1}{3 a^{2} \sqrt{a}}$ by $\frac{\sqrt{a}}{\sqrt{a}}$ .

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

Move $\sqrt{a}$ .

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

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

$\frac{\sqrt{a}}{3 a^{2} ({\sqrt{a}}^{1} \sqrt{a})}$

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

$\frac{\sqrt{a}}{3 a^{2} ({\sqrt{a}}^{1} {\sqrt{a}}^{1})}$

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

$\frac{\sqrt{a}}{3 a^{2} {\sqrt{a}}^{1 + 1}}$

Add $1$ and $1$ .

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

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

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

$\frac{\sqrt{a}}{3 a^{2} {(a^{\frac{1}{2}})}^{2}}$

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

$\frac{\sqrt{a}}{3 a^{2} a^{\frac{1}{2} \cdot 2}}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{\sqrt{a}}{3 a^{2} a^{1}}$

Simplify.

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

Step 5

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

Move $a$ .

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

Multiply $a$ by $a^{2}$ .

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

$\frac{\sqrt{a}}{3 (a^{1} a^{2})}$

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

$\frac{\sqrt{a}}{3 a^{1 + 2}}$

Add $1$ and $2$ .

$\frac{\sqrt{a}}{3 a^{3}}$

Simplify (2a)/( square root of 36a^7)

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