Question

What does the quotient rule of exponents say?

Answer 1

See explanation

Explanation:

The quotient rule of exponents is:

`(a^m)/(a^n)=a^(m-n); a!=0`

What this means that if you are dealing with a quotient problem involving exponents, then you can subtract the exponents if the bases are the same.

This is how it works. Lets consider the following:

`(a^7)/(a^3)`

If we write out the exponents like so...

`(a^7)/(a^3)=(a*a*a*a*a*a*a)/(a*a*a)`

... we can cancel out some of the `a`'s because `a/a=1`

`:. a^7/a^3=(a*a*a*a*color(red)cancel(a)*color(red)cancela*color(red)cancela)/(color(red)cancela*color(red)cancela*color(red)cancela)=(a*a*a*a)/1=a^4`

This is essentially subtraction and rather then writing out every term we can subtract the exponents like so:

`a^7/a^3=a^(7-3)=a^4`

Remember: You can only apply this rule if the bases are the same!

For example we could not simply `a^4/b^2` much further because the bases are not the same so there aren't any like terms to combine.

Lets look at a final example:

Simplify:

`(a^13*b^3*c^5)/(a^5*b^2*c^2)`

Here we have three different bases so we can treat this as three small problems if we split up the problem such that:

`(a^13*b^3*c^5)/(a^5*b^2*c^2)=color(blue)(a^13/a^5)*color(red)(b^3/b^2)*color(green)(c^5/c^2)`

We now apply the rule:

`color(blue)(a^(13-5))*color(red)(b^(3-2))*color(green)(c^(5-2))= color(blue)(a^(8))*color(red)(b^(1))*color(green)(c^(3))= a^8bc^3`

I hope this helped. :)