Simplify square root of 36/7* square root of 3/77

$\sqrt{\frac{36}{7}} \cdot \sqrt{\frac{3}{77}}$

Step 1

Combine using the product rule for radicals.

$\sqrt{\frac{36}{7} \cdot \frac{3}{77}}$

Step 2

Multiply $\frac{36}{7}$ by $\frac{3}{77}$ .

$\sqrt{\frac{36 \cdot 3}{7 \cdot 77}}$

Step 3

Multiply.

Multiply $36$ by $3$ .

$\sqrt{\frac{108}{7 \cdot 77}}$

Multiply $7$ by $77$ .

$\sqrt{\frac{108}{539}}$

Step 4

Rewrite $\sqrt{\frac{108}{539}}$ as $\frac{\sqrt{108}}{\sqrt{539}}$ .

$\frac{\sqrt{108}}{\sqrt{539}}$

Step 5

Simplify the numerator.

Rewrite $108$ as $6^{2} \cdot 3$ .

Factor $36$ out of $108$ .

$\frac{\sqrt{36 (3)}}{\sqrt{539}}$

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

$\frac{\sqrt{6^{2} \cdot 3}}{\sqrt{539}}$

Pull terms out from under the radical.

$\frac{6 \sqrt{3}}{\sqrt{539}}$

Step 6

Simplify the denominator.

Rewrite $539$ as $7^{2} \cdot 11$ .

Factor $49$ out of $539$ .

$\frac{6 \sqrt{3}}{\sqrt{49 (11)}}$

Rewrite $49$ as $7^{2}$ .

$\frac{6 \sqrt{3}}{\sqrt{7^{2} \cdot 11}}$

Pull terms out from under the radical.

$\frac{6 \sqrt{3}}{7 \sqrt{11}}$

Step 7

Multiply $\frac{6 \sqrt{3}}{7 \sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{6 \sqrt{3}}{7 \sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}}$

Step 8

Combine and simplify the denominator.

Multiply $\frac{6 \sqrt{3}}{7 \sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{6 \sqrt{3} \sqrt{11}}{7 \sqrt{11} \sqrt{11}}$

Move $\sqrt{11}$ .

$\frac{6 \sqrt{3} \sqrt{11}}{7 (\sqrt{11} \sqrt{11})}$

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{6 \sqrt{3} \sqrt{11}}{7 ({\sqrt{11}}^{1} \sqrt{11})}$

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{6 \sqrt{3} \sqrt{11}}{7 ({\sqrt{11}}^{1} {\sqrt{11}}^{1})}$

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

$\frac{6 \sqrt{3} \sqrt{11}}{7 {\sqrt{11}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{6 \sqrt{3} \sqrt{11}}{7 {\sqrt{11}}^{2}}$

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\frac{6 \sqrt{3} \sqrt{11}}{7 {(11^{\frac{1}{2}})}^{2}}$

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

$\frac{6 \sqrt{3} \sqrt{11}}{7 \cdot 11^{\frac{1}{2} \cdot 2}}$

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

$\frac{6 \sqrt{3} \sqrt{11}}{7 \cdot 11^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{6 \sqrt{3} \sqrt{11}}{7 \cdot 11^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{6 \sqrt{3} \sqrt{11}}{7 \cdot 11^{1}}$

Evaluate the exponent.

$\frac{6 \sqrt{3} \sqrt{11}}{7 \cdot 11}$

Step 9

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{6 \sqrt{11 \cdot 3}}{7 \cdot 11}$

Multiply $11$ by $3$ .

$\frac{6 \sqrt{33}}{7 \cdot 11}$

Step 10

Multiply $7$ by $11$ .

$\frac{6 \sqrt{33}}{77}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{6 \sqrt{33}}{77}$

Decimal Form:

$0.44762825 \dots$

Simplify square root of 36/7* square root of 3/77

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