Question

Please check my homework on Negative Indices!

Answer 1

Our first expression is $(2p^{-3} q^{8} )^{4}$. Upon distributing the exponent 4 on all the terms, we get:

$(2p^{-3} q^{8} )^{4}=2^{4}(p^{-3})^{4}(q^{8})^{4}=16p^{-12}q^{32}=\frac{16q^{32}}{p^{12}}$

Therefore, your answer is correct for this part. :)

Second expression is $(2m^{-4}n^{5})^{-2}$. Upon distributing the exponent -2 on all the terms, we get:

$(2m^{-4}n^{5})^{-2}=2^{-2}(m^{-4})^{-2}(n^{5})^{-2}=2^{-2}m^{8}n^{-10}=\frac{m^{8}}{4n^{10}}$

Your second answer is correct too.

Our third expression is $2(5g^{-4}h^{-6})^{2}$. Upon distributing the exponent 2 on all the terms, we get:

$2(5g^{-4}h^{-6})^{2}=2(5^{2})(g^{-4})^{2}(h^{-6})^{2}=2(25)g^{-8}h^{-12}=\frac{50}{g^{8}h^{12}}$

This one is not correct. Your answer would have been correct, if the exponent were -2 instead of 2 in this part.

Our forth and last expression is $4(2c^{-3}d^{6})^{-5}$. Upon distributing the exponent -5 on all the terms inside the parenthesis, we get:

$4(2c^{-3}d^{6})^{-5}=4(2^{-5})(c^{-3})^{-5}(d^{6})^{-5}=\frac{4}{2^{5}}(c^{15})(d^{-30})=\frac{4c^{15}}{32d^{30}}=\frac{c^{15}}{8d^{30}}$

Therefore, your answer for this part is also correct.

Looking at your work, I don't think you made a mistake in number 3 also, probably mis-typed the question while writing here :)