# Index Laws Homework Hotline

## Index laws

**Multiplying and dividing**

When multiplying you add the indices, and when dividing you subtract the indices.

So it follows that:

p^{3} × p^{7} = p^{10}, and s^{5} ÷ s^{3} = s^{2}

For the expression:

4s^{3} x 3s^{2}

The numbers in front of the variables follow the usual rules of multiplication and division, but index numbers follow the rules of indices. So we multiply 4 and 3 and add 3 and 2

4s^{3} × 3s^{2} = 12s^{5}

- Question
What is 3c

^{2}× 5c^{4}?

- Answer
**To work it out:**- Add the indices:
- 2 + 4 = 6

- Multiply the numbers in front of the variable:
- 3 x 5 = 15

**Note:** Take care when multiplying and dividing expressions such as y × y^{4} or z^{3} ÷ z.

y is the same as y^{1}, so y × y^{4} = y^{5}.

z is the same as z^{1}, so z^{3} ÷ z = z^{2}.

## Adding and subtracting

You can only add and subtract 'like terms'.

3, 4 and 20 are all like terms (because they are all numbers).

a, 3a and 200a are all like terms (because they are all multiples of a).

a^{2}, 10a^{2} and -2a^{2} are all like terms (because they are all multiples of a^{2})

You cannot simplify an expression like 4p + p^{2} because 4p and p^{2} are not like terms.

But you can simplify 3r^{2} + 5r^{2} + r^{2}.

3r^{2} + 5r^{2} + r^{2} tells us that we have 'three lots of r^{2}' + 'five lots of r^{2}' + 'one lot of r^{2}' - so in total 'nine lots of r^{2}', or 9r^{2}.

So, 3r^{2} + 5r^{2} + r ^{2} = 9r^{2}

- Question
What is s

^{2}+ 8s^{2}- 2s^{2}?

- Answer
**Answer:**7s^{2}Remember that 1 + 8 - 2 = 7, so s

^{2}+ 8s^{2}- 2s^{2}= 7s^{2}

Remember that if we have a mix of terms we must gather like terms before we simplify.

**Example**

3p^{2} + 2p + 4 - 2p^{2} + 5 = 3p^{2} - 2p^{2} + 2p + 4 + 5 = p^{2} + 2p + 9

Back to Algebra index

## Index laws

## Multiplication

How can we work out 2^{3} × 2^{5}?

2^{3} = 2 × 2 × 2

2^{5} = 2 × 2 × 2 × 2 × 2

so 2^{3} × 2^{5} = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2^{8}

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

In general, 2^{m} × 2^{n} =2^{(m + n)}

### Examples

2^{5} × 2^{4} = 2^{(5 + 4)} = 2^{9}

2^{7} × 2^{3} = 2^{(7 + 3)} = 2^{10}

The rule also works for other numbers, so

3^{4} × 3^{2} = 3^{(4 + 2)} = 3^{6}

25^{6} × 25^{4} = 25^{(6 + 4)} = 25^{10}

## Division

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

So 2^{5} ÷ 2^{3} = 2^{2}

In general, 2^{m} ÷ 2^{n} = 2^{(m - n)}

### Example

2^{5} ÷ 2^{2} = 2^{(5 - 2)} = 2^{3}

2^{7} ÷ 2^{3} = 2^{(7 - 3)} = 2^{4}

The rule also works for other numbers, so

5^{10} ÷ 5^{3} =5^{(10 - 3)} = 5^{7}

45^{9} ÷ 45^{4} = 45^{(9 - 4)} = 45^{5}

### Dividing numbers with roots or powers

Five 2s are divided by three 2s

More from Number

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