Use the properties of exponents to simplify the quantity negative 8 m to the fifth n to the eighth over 2 m to the seventh n to the negative third, all raised to the negative fourth power. Write the answer with positive exponents.

Step-by-step solution

1. 1

   Simplify inside the parentheses before touching the outer exponent. Doing the division first keeps the base as small as possible, so the outer power is applied to three simple factors instead of a fraction of six.

2. 2

   Divide the coefficients and subtract the exponents. For like bases, $\dfrac{a^m}{a^n}=a^{m-n}$:

   $\frac{-8}{2}=-4,\qquad m^{5-7}=m^{-2},\qquad n^{8-(-3)}=n^{11}$

   Note the double negative on $n$: subtracting $-3$ adds $3$. The bracket is now $\left(-4m^{-2}n^{11}\right)^{-4}$.

3. 3

   Distribute the outer exponent over each factor. Using $(abc)^k=a^kb^kc^k$ and $(a^m)^k=a^{mk}$:

   $(-4)^{-4}\left(m^{-2}\right)^{-4}\left(n^{11}\right)^{-4} = (-4)^{-4}\,m^{8}\,n^{-44}$

4. 4

   Evaluate the numeric factor and watch the sign. $-4$ is raised to an even power, so the result is positive:

   $(-4)^{-4}=\frac{1}{(-4)^4}=\frac{1}{256}$

   If the outer exponent had been odd, this factor would have been negative instead.

5. 5

   Rewrite every negative exponent as a reciprocal. $n^{-44}=\dfrac{1}{n^{44}}$, so the pieces combine to

   $\frac{m^{8}}{256\,n^{44}}$

6. 6

   Check with a quick substitution. With $m=1.7$ and $n=1.3$, the original expression evaluates to $2.6415\times10^{-6}$, and $\dfrac{1.7^{8}}{256\cdot 1.3^{44}}$ gives the same $2.6415\times10^{-6}$.