Index laws for multiplication and division

Multiplication

How can we work out 2^3 \times 2^5?

2^3 = 2 \times 2 \times 2

2^5 = 2 \times 2 \times 2 \times 2 \times 2

so 2^3 \times 2^5 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^8

There are {3} twos from 2^3 and {5} twos from 2^5, so altogether there are {8} twos.

The index {8} can be found from adding the indices {3} and {5}. Remember, this only works when we are multiplying powers of the same number together.

In general: 2^m \times 2^n =2^{(m + n)}

Examples

2^5 \times 2^4 = 2^{(5 + 4)} = 2^9

2^7 \times 2^3 = 2^{(7 + 3)} = 2^{10}

The rule also works for other numbers, so:

3^4 \times 3^2 = 3^{(4 + 2)} = 3^6

15^6 \times 15^4 = 15^{(6 + 4)} = 15^{10}

Division

If you divide 2^5 by 2^3 you see that some of the {2}s cancel:

Dividing powers diagram

Dividing numbers with powers

Five 2s are divided by three 2s

So 2^5 \div 2^3 = 2^2.

There are {5} twos from {2}^{5} and {3} twos from {2}^{3}, so after dividing there are {2} twos. The index {2} can be found from subtracting the indices, 5 - 3 = 2. Remember, this only works when we are dividing powers of the same number.

In general, 2^m \div 2^n =2^{(m - n)}.

Example

2^5 \div 2^2 = 2^{(5 - 2)} = 2^3

2^7 \div 2^3 = 2^{(7 - 3)} = 2^4

The rule also works for other numbers:

{5}^{10}\div{5}^{3} ={5}^{({10} - {3})} = {5}^{7}

45^9 \div 45^4 = 45^{(9 - 4)} = 45^5

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