Multiplying and Dividing Power Terms

## Multiplying and Dividing Power Terms

The Laws of Indices which apply to numbers are also applied to algebra. Consider y^3 xx y^2.

The term y^3 can be expanded to y xx y xx y. The other term, y^2, can be expanded to y xx y. Therefore y^3 xx y^2 = y xx y xx y xx y xx y, which is equal to y^5.

This can be written as y^a xx y^b = y^(a + b). When multiplying the same variable, add the powers. Note that the variable letter must be the same for both the terms being multiplied.

Division works in a similar way. The rule is written as y^a ÷ y^b = y^(a - b). When dividing the same variable, subtract the powers. The example e^6 ÷ e^3 can be shown as:

frac(e^6)(e^2)

= frac(e xx e xx e xx e xx e xx e)(e xx e)

= frac(e xx e xx e xx e)(1)

= e^4

The division rule leads to the fact that e^0 = 1. This can be shown by e^2 ÷ e^2

= frac(e xx e)(e xx e)

= frac(e)(e) (divided top and bottom by e)

= 1 (divided top and bottom by e again)

which leads to the fact that any value to the power of zero equals 1.

## Example 1

Simplify x^3 xx 4x^3

x^3 xx 4x^3

= 4 xx x^(3 + 3)

= 4x^6

Answer: 4x^6

## Example 2

Simplify 4a^5b ÷ 2a^3

Calculate the numbers: 4 ÷ 2 = 2

Calculate the variable a: a^5 ÷ a^3 = a^(5-3) = a^2

The variable b is not divided

Put it all together: 2a^2b

Answer: 2a^2b