Question

How do you simplify `g^ { 5} \cdot g ^ { - 7} \cdot g ^ { - 2}`?

Answer 1

See a solution process below:

Explanation:

Use this rule of exponents to multiply the terms:

`x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) + color(blue)(b))`

`g^color(red)(5) * g^color(blue)(-7) * g^color(green)(-2) => g^(color(red)(5)+color(blue)(-7)+color(green)(-2)) => g^-4`

If it is necessary to eliminate negative exponents we can use this rule of exponents:

`x^color(red)(a) = 1/x^color(red)(-a)`

`g^color(red)(-4) => 1/(g^color(red)(- -4)) => 1/g^4`

Answer 2

`1/g^4`

Answers should usually be given with positive indices.

Explanation:

Let's use one of the laws f indices to get rid of the negative indices:

`x^-m = 1/x^m`

`g^5 * color(blue)(g^-7*g^-2)`

`= g^5/(color(blue)(g^7 xxg^2))`

`= g^5/g^9`

When dividing you subtract the indices of like bases.
However in this case the bigger index is in the denominator.
Note the following:

`x^m/x^n =x^(m-n)" "`if `m>n" "` gives a positive index

`x^m/x^n = 1/(x^(n-m))" "` if `n>m" "` gives a positive index

`g^5/(g^9) = 1/g^4`