Simplify square root of (147m^5n^7)/(441z^6)

$\sqrt{\frac{147 m^{5} n^{7}}{441 z^{6}}}$

Step 1

Reduce the expression $\frac{147 m^{5} n^{7}}{441 z^{6}}$ by cancelling the common factors.

Factor $147$ out of $147 m^{5} n^{7}$ .

$\sqrt{\frac{147 (m^{5} n^{7})}{441 z^{6}}}$

Factor $147$ out of $441 z^{6}$ .

$\sqrt{\frac{147 (m^{5} n^{7})}{147 (3 z^{6})}}$

Cancel the common factor.

$\sqrt{\frac{147 (m^{5} n^{7})}{147 (3 z^{6})}}$

Rewrite the expression.

$\sqrt{\frac{m^{5} n^{7}}{3 z^{6}}}$

Step 2

Rewrite $\frac{m^{5} n^{7}}{3 z^{6}}$ as ${(\frac{m^{2} n^{3}}{z^{3}})}^{2} \frac{m n}{3}$ .

Factor the perfect power ${(m^{2} n^{3})}^{2}$ out of $m^{5} n^{7}$ .

$\sqrt{\frac{{(m^{2} n^{3})}^{2} (m n)}{3 z^{6}}}$

Factor the perfect power ${(z^{3})}^{2}$ out of $3 z^{6}$ .

$\sqrt{\frac{{(m^{2} n^{3})}^{2} (m n)}{{(z^{3})}^{2} \cdot 3}}$

Rearrange the fraction $\frac{{(m^{2} n^{3})}^{2} (m n)}{{(z^{3})}^{2} \cdot 3}$ .

$\sqrt{{(\frac{m^{2} n^{3}}{z^{3}})}^{2} \frac{m n}{3}}$

Step 3

Pull terms out from under the radical.

$\frac{m^{2} n^{3}}{z^{3}} \sqrt{\frac{m n}{3}}$

Step 4

Rewrite $\sqrt{\frac{m n}{3}}$ as $\frac{\sqrt{m n}}{\sqrt{3}}$ .

$\frac{m^{2} n^{3}}{z^{3}} \cdot \frac{\sqrt{m n}}{\sqrt{3}}$

Step 5

Combine.

$\frac{m^{2} n^{3} \sqrt{m n}}{z^{3} \sqrt{3}}$

Step 6

Multiply $\frac{m^{2} n^{3} \sqrt{m n}}{z^{3} \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{m^{2} n^{3} \sqrt{m n}}{z^{3} \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 7

Combine and simplify the denominator.

Multiply $\frac{m^{2} n^{3} \sqrt{m n}}{z^{3} \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{m^{2} n^{3} \sqrt{m n} \sqrt{3}}{z^{3} \sqrt{3} \sqrt{3}}$

Move $\sqrt{3}$ .

$\frac{m^{2} n^{3} \sqrt{m n} \sqrt{3}}{z^{3} (\sqrt{3} \sqrt{3})}$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{m^{2} n^{3} \sqrt{m n} \sqrt{3}}{z^{3} ({\sqrt{3}}^{1} \sqrt{3})}$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{m^{2} n^{3} \sqrt{m n} \sqrt{3}}{z^{3} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

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

$\frac{m^{2} n^{3} \sqrt{m n} \sqrt{3}}{z^{3} {\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{m^{2} n^{3} \sqrt{m n} \sqrt{3}}{z^{3} {\sqrt{3}}^{2}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{m^{2} n^{3} \sqrt{m n} \sqrt{3}}{z^{3} {(3^{\frac{1}{2}})}^{2}}$

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

$\frac{m^{2} n^{3} \sqrt{m n} \sqrt{3}}{z^{3} \cdot 3^{\frac{1}{2} \cdot 2}}$

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

$\frac{m^{2} n^{3} \sqrt{m n} \sqrt{3}}{z^{3} \cdot 3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{m^{2} n^{3} \sqrt{m n} \sqrt{3}}{z^{3} \cdot 3^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{m^{2} n^{3} \sqrt{m n} \sqrt{3}}{z^{3} \cdot 3^{1}}$

Evaluate the exponent.

$\frac{m^{2} n^{3} \sqrt{m n} \sqrt{3}}{z^{3} \cdot 3}$

Step 8

Combine using the product rule for radicals.

$\frac{m^{2} n^{3} \sqrt{3 m n}}{z^{3} \cdot 3}$

Step 9

Move $3$ to the left of $z^{3}$ .

$\frac{m^{2} n^{3} \sqrt{3 m n}}{3 z^{3}}$

Simplify square root of (147m^5n^7)/(441z^6)

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