# Laws of Exponents - Multiplying Powers with the Same Base

## Formula

For any non-zero integer a, where m and n are whole numbers, am × an =am + n.

## Notes

### Multiplying Powers With the Same Base:

For any non-zero integer a, where m and n are whole numbers, am × an =am + n.

(i) Let us calculate 42 × 43.

= 42 × 43

= (4 × 4) × (4 × 4 × 4)

= 4 × 4 × 4 × 4 × 4 = 45 = 42+3...............(am × an = am+n)

Note that the base in 42 and 43 is the same and the sum of the exponents, i.e., 2 and 3 is 5.

(ii) (– 11)4 × (– 11)3

= [(– 11) × (– 11) × (– 11) × (– 11)] × [(– 11) × (– 11) × (– 11)]

= (– 11) × (– 11) × (– 11) × (– 11) × (– 11) × (– 11) × (– 11) = (– 11)7 = (– 11)4+3........(am × an = am+n)

Again, note that the base is the same and the sum of exponents, i.e., 4 and 3, is 7.

(iii) 32 × 34 × 38

= 32 × 34 × 38 = (3)2 + 4 + 8.............(am × an = am+n)

= 314.

## Example

Solve: (-3)2 × (-3)3.

(-3)2 × (-3)3

= (-3) × (-3) × (-3) × (-3) × (-3)

= (-3)5

Therefore, (-3)2 × (-3)3 = (-3)2 + 3 = (-3)5.

## Example

Solve: ((-2)/5)^2 xx ((-2)/5)^3

((-2)/5)^2 xx ((-2)/5)^3

= ((-2)/5) xx ((-2)/5) xx ((-2)/5) xx ((-2)/5) xx ((-2)/5)

= ((-2)/5)^5

Therefore, ((-2)/5)^2 xx ((-2)/5)^3 = ((-2)/5)^(2 + 3) = ((-2)/5)^5.

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