Question

Which expression is equivalent to ( 4mn / m^-2n^6)^-2

a. n^6 / 16m^8
b. n^10 / 16m^6
c. n^10 / 8m^8
d. 4m^3 / n^8

Answer 1

Here are a few rules with exponents that apply here:

1. Raising a power to a power:
   `(x^m)^n=x^(m*n)`
2. Dividing exponents of the same base:
   `(x^m)/(x^n)=x^(m-n)`
3. Converting negative exponents to positive ones:
   `x^(-m)=(1)/(x^m)\ \textsf{and}\ (1)/(x^(-m))=x^m`

Firstly, solve the outer exponent:

`((4mn)/(m^(-2)n^(6)))^(-2)=(4^(-2)m^(-2)n^(-2))/(m^(-2*-2)n^(6*-2))=(4^(-2)m^(-2)n^(-2))/(m^4n^(-12))`

Next, divide:

`(4^(-2)m^(-2)n^(-2))/(m^4n^(-12))=4^(-2)m^(-2-4)n^(-2-(-12))=4^(-2)m^(-6)n^(10)`

Next, convert the negative exponents:

`4^(-2)m^(-6)n^(10)=(n^(10))/(4^2m^6)=(n^(10))/(16m^6)`

Your final answer is n^10/16m^6 , or B.