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

Question

Ethan Adkins · Accepted Answer

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

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)#

Isaiah Anthony · Answer

undefined To simplify the expression (3a^-1)^-1 (9a^2 b^3)^-2, we first apply the negative exponent rule to rewrite the expressions as fractions. Then, we multiply the fractions together and simplify the result.

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

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

Multiplying the fractions together, we get:

3a^1 * (81/a^4 b^6)

Simplifying the expression further, we combine like terms in the numerator and denominator:

(3 * 81) * (a^1 / a^4) * (1 / b^6) = 243 / (a^3 b^6)