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

2 Answers
Jan 30, 2018

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#

Jan 30, 2018

#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#