Question

How do you simplify `((a^3b^4)/(a^2b))^3` using the exponential properties?

Answer 1

In order to simplify this term involving an exponent on the outside of the parentheses and exponents inside the parentheses, there are two basic rules to follow. First multiply the exponent outside the parentheses times the exponents inside the parentheses according to the exponent rule `(x^n)^m` = `x^(n·m)`. Since this problem involves division with exponents, the exponents in the denominator are then subtracted from the exponents in the numerator according to the exponent rule `x^n/x^m` = `x^(n-m)`. You may only subtract exponents that have the same base, such as `x` or `y`.

`((a^3b^4)/(a^2b))^3`

First multiply all exponents inside the parentheses times the exponent outside the parentheses.

`((a^(3·3)b^(4·3))/(a^(2·3)b^(1·3)))` = `((a^9b^12)/(a^6b^3))`

Next subtract the exponents in the denominator from the exponents in the numerator that have the same base. Now the term has been simplified.

`((a^9b^12)/(a^6b^3))` = `(a^(9-6)b^(12-3))` = `a^3b^9`