Divide Indices
for 8th grade

Divide indices with the same base by subtracting powers

What are indices?

Indices are terms that have been raised to a power such as 5⁶ or y⁴. This means the term has been multiplied by itself that number of times.

6⁴ = 6 x 6 x 6 x 6     and     y⁵ = y x y x y x y x y

So the term indice simply means a number or term raised to a power.

How to divide indices with the same base

Indices with the same base can be divided by subtracting powers. So a³ and a⁶ have the same base "a", but x³ and y⁶ do not have the same base and cannot be simplified by subtracting powers.

To understand why dividing indices with the same base involves subtracting powers, consider x⁷ ÷ x⁴. If we expand both indices we get:

x x x x x x x x x x x x x
x x x x x x x

Since we know that x divided by x is 1, we can cancel terms top and bottom to simplify the expression leaving x x x x x.

x⁷ ÷ x⁴ = x³

So we can see that dividing x⁷ by x⁴ gives us x³. In general, you can divide indices with the same base by subtracting the powers.

If one or more of the terms is raised to a negative power, you should follow simple arithmetic rules to subtract the powers.

p⁻⁵ ÷ p³ = p⁻⁸

p⁻⁵ ÷ p⁻³ = p⁻²

If there are coefficients in front of the bases, divide the coefficients first and then subtract the powers.

15p⁵ ÷ 3p³ = 5p⁵