Question

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

Answer 1

`1/243 * 1/a^3 * 1/b^6`

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 `a` by using the 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)`