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
#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)#