# How do you simplify #(3a^-1)^-1 (9a^2 b^3)^-2#?

##### 1 Answer

#### Explanation:

Start by recognizing that a *negative exponent* can be written as

#n ^(-a) = 1/n^a#

Now, notice that you can rewrite the first term of the expression as

#(3a^(-1))^(-1) = 3^(-1) * (a^(-1))^(-1) = 1/3^1 * (a^(-1))^(-1)#

You can actually bypass the negative exponent for *power of a power property* of exponents

#color(blue)( (n^a)^b = n^(a * b))#

In this case, you have

#(3a^(-1))^(-1) = 1/3 * a^( (-1) * (-1)) = 1/3 * a^1 = 1/3 *a#

The second term of the expression will be

#(9a^2b^3)^(-2) = 1/(9a^2b^3)^2 = 1/9^2 * 1/(a^2)^2 * 1/(b^3)^2#

#= 1/81 * 1/a^4 * 1/b^6#

This means that you have

#(3a^(-1))^(-1) * (9a^2b^3)^(-2) = 1/3 * a * 1/81 * 1/a^4 * 1/b^6#

This can be simplifed to

#1/3 * 1/81 * 1/a^3 * 1/b^6 = color(green)(1/243 * 1/a^3 * 1/b^6)#