Simplify ((4m^-2n^8)/(9m^-6n^-8))^(1/2)

${(\frac{4 m^{- 2} n^{8}}{9 m^{- 6} n^{- 8}})}^{\frac{1}{2}}$

Step 1

Move $m^{- 6}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{4 m^{- 2} n^{8} m^{6}}{9 n^{- 8}})}^{\frac{1}{2}}$

Step 2

Move $n^{- 8}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{4 m^{- 2} n^{8} m^{6} n^{8}}{9})}^{\frac{1}{2}}$

Step 3

Simplify the numerator.

Multiply $m^{- 2}$ by $m^{6}$ by adding the exponents.

Move $m^{6}$ .

${(\frac{4 (m^{6} m^{- 2}) n^{8} n^{8}}{9})}^{\frac{1}{2}}$

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

${(\frac{4 m^{6 - 2} n^{8} n^{8}}{9})}^{\frac{1}{2}}$

Subtract $2$ from $6$ .

${(\frac{4 m^{4} n^{8} n^{8}}{9})}^{\frac{1}{2}}$

Multiply $n^{8}$ by $n^{8}$ by adding the exponents.

Move $n^{8}$ .

${(\frac{4 m^{4} (n^{8} n^{8})}{9})}^{\frac{1}{2}}$

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

${(\frac{4 m^{4} n^{8 + 8}}{9})}^{\frac{1}{2}}$

Add $8$ and $8$ .

${(\frac{4 m^{4} n^{16}}{9})}^{\frac{1}{2}}$

Step 4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{4 m^{4} n^{16}}{9}$ .

$\frac{{(4 m^{4} n^{16})}^{\frac{1}{2}}}{9^{\frac{1}{2}}}$

Apply the product rule to $4 m^{4} n^{16}$ .

$\frac{{(4 m^{4})}^{\frac{1}{2}} {(n^{16})}^{\frac{1}{2}}}{9^{\frac{1}{2}}}$

Apply the product rule to $4 m^{4}$ .

$\frac{4^{\frac{1}{2}} {(m^{4})}^{\frac{1}{2}} {(n^{16})}^{\frac{1}{2}}}{9^{\frac{1}{2}}}$

Step 5

Simplify the numerator.

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

$\frac{{(2^{2})}^{\frac{1}{2}} {(m^{4})}^{\frac{1}{2}} {(n^{16})}^{\frac{1}{2}}}{9^{\frac{1}{2}}}$

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

$\frac{2^{2 (\frac{1}{2})} {(m^{4})}^{\frac{1}{2}} {(n^{16})}^{\frac{1}{2}}}{9^{\frac{1}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2^{2 (\frac{1}{2})} {(m^{4})}^{\frac{1}{2}} {(n^{16})}^{\frac{1}{2}}}{9^{\frac{1}{2}}}$

Rewrite the expression.

$\frac{2^{1} {(m^{4})}^{\frac{1}{2}} {(n^{16})}^{\frac{1}{2}}}{9^{\frac{1}{2}}}$

Evaluate the exponent.

$\frac{2 {(m^{4})}^{\frac{1}{2}} {(n^{16})}^{\frac{1}{2}}}{9^{\frac{1}{2}}}$

Multiply the exponents in ${(m^{4})}^{\frac{1}{2}}$ .

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

$\frac{2 m^{4 (\frac{1}{2})} {(n^{16})}^{\frac{1}{2}}}{9^{\frac{1}{2}}}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$\frac{2 m^{2 (2) \frac{1}{2}} {(n^{16})}^{\frac{1}{2}}}{9^{\frac{1}{2}}}$

Cancel the common factor.

$\frac{2 m^{2 \cdot 2 \frac{1}{2}} {(n^{16})}^{\frac{1}{2}}}{9^{\frac{1}{2}}}$

Rewrite the expression.

$\frac{2 m^{2} {(n^{16})}^{\frac{1}{2}}}{9^{\frac{1}{2}}}$

Multiply the exponents in ${(n^{16})}^{\frac{1}{2}}$ .

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

$\frac{2 m^{2} n^{16 (\frac{1}{2})}}{9^{\frac{1}{2}}}$

Cancel the common factor of $2$ .

Factor $2$ out of $16$ .

$\frac{2 m^{2} n^{2 (8) \frac{1}{2}}}{9^{\frac{1}{2}}}$

Cancel the common factor.

$\frac{2 m^{2} n^{2 \cdot 8 \frac{1}{2}}}{9^{\frac{1}{2}}}$

Rewrite the expression.

$\frac{2 m^{2} n^{8}}{9^{\frac{1}{2}}}$

Step 6

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

$\frac{2 m^{2} n^{8}}{{(3^{2})}^{\frac{1}{2}}}$

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

$\frac{2 m^{2} n^{8}}{3^{2 (\frac{1}{2})}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 m^{2} n^{8}}{3^{2 (\frac{1}{2})}}$

Rewrite the expression.

$\frac{2 m^{2} n^{8}}{3^{1}}$

Evaluate the exponent.

$\frac{2 m^{2} n^{8}}{3}$

Simplify ((4m^-2n^8)/(9m^-6n^-8))^(1/2)

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