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

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

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Combine $\frac{1}{m}$ and $n$ .

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

Apply the product rule to $\frac{n}{m}$ .

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

Combine.

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

Multiply $n^{\frac{1}{2}}$ by $1$ .

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

Multiply $m^{9}$ by $m^{\frac{1}{2}}$ by adding the exponents.

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

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

To write $9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Combine $9$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $9$ by $2$ .

$\frac{n^{\frac{1}{2}}}{m^{\frac{18 + 1}{2}}} n^{8}$

Add $18$ and $1$ .

$\frac{n^{\frac{1}{2}}}{m^{\frac{19}{2}}} n^{8}$

Multiply $\frac{n^{\frac{1}{2}}}{m^{\frac{19}{2}}} n^{8}$ .

Combine $\frac{n^{\frac{1}{2}}}{m^{\frac{19}{2}}}$ and $n^{8}$ .

$\frac{n^{\frac{1}{2}} n^{8}}{m^{\frac{19}{2}}}$

Multiply $n^{\frac{1}{2}}$ by $n^{8}$ by adding the exponents.

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

$\frac{n^{\frac{1}{2} + 8}}{m^{\frac{19}{2}}}$

To write $8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{n^{\frac{1}{2} + 8 \cdot \frac{2}{2}}}{m^{\frac{19}{2}}}$

Combine $8$ and $\frac{2}{2}$ .

$\frac{n^{\frac{1}{2} + \frac{8 \cdot 2}{2}}}{m^{\frac{19}{2}}}$

Combine the numerators over the common denominator.

$\frac{n^{\frac{1 + 8 \cdot 2}{2}}}{m^{\frac{19}{2}}}$

Simplify the numerator.

Multiply $8$ by $2$ .

$\frac{n^{\frac{1 + 16}{2}}}{m^{\frac{19}{2}}}$

Add $1$ and $16$ .

$\frac{n^{\frac{17}{2}}}{m^{\frac{19}{2}}}$