Simplify ( square root of 3a)/( square root of 27a^7)

$\frac{\sqrt{3 a}}{\sqrt{27 a^{7}}}$

Step 1

Combine $\sqrt{3 a}$ and $\sqrt{27 a^{7}}$ into a single radical.

$\sqrt{\frac{3 a}{27 a^{7}}}$

Step 2

Reduce the expression $\frac{3 a}{27 a^{7}}$ by cancelling the common factors.

Factor $3$ out of $3 a$ .

$\sqrt{\frac{3 (a)}{27 a^{7}}}$

Factor $3$ out of $27 a^{7}$ .

$\sqrt{\frac{3 (a)}{3 (9 a^{7})}}$

Cancel the common factor.

$\sqrt{\frac{3 a}{3 (9 a^{7})}}$

Rewrite the expression.

$\sqrt{\frac{a}{9 a^{7}}}$

Step 3

Cancel the common factor of $a$ and $a^{7}$ .

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

$\sqrt{\frac{a^{1}}{9 a^{7}}}$

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

$\sqrt{\frac{a \cdot 1}{9 a^{7}}}$

Cancel the common factors.

Factor $a$ out of $9 a^{7}$ .

$\sqrt{\frac{a \cdot 1}{a (9 a^{6})}}$

Cancel the common factor.

$\sqrt{\frac{a \cdot 1}{a (9 a^{6})}}$

Rewrite the expression.

$\sqrt{\frac{1}{9 a^{6}}}$

Step 4

Rewrite $1$ as $1^{2}$ .

$\sqrt{\frac{1^{2}}{9 a^{6}}}$

Step 5

Rewrite $9 a^{6}$ as ${(3 a^{3})}^{2}$ .

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

Step 6

Rewrite $\frac{1^{2}}{{(3 a^{3})}^{2}}$ as ${(\frac{1}{3 a^{3}})}^{2}$ .

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

Step 7

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

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