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:

p3 × p7 = p10, and s5 ÷ s3 = s2

For the expression:

4s3 x 3s2

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

4s3 × 3s2 = 12s5

Question

What is 3c2 × 5c4?

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 × y4 or z3 ÷ z.

y is the same as y1, so y × y4 = y5.

z is the same as z1, so z3 ÷ z = z2.

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).

a2, 10a2 and -2a2 are all like terms (because they are all multiples of a2)

You cannot simplify an expression like 4p + p2 because 4p and p2 are not like terms.

But you can simplify 3r2 + 5r2 + r2.

3r2 + 5r2 + r2 tells us that we have 'three lots of r2' + 'five lots of r2' + 'one lot of r2' - so in total 'nine lots of r2', or 9r2.

So, 3r2 + 5r2 + r 2 = 9r2

Question

What is s2 + 8s2 - 2s2?

Answer

Answer: 7s2

Remember that 1 + 8 - 2 = 7, so s2 + 8s2 - 2s2 = 7s2

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

Example

3p2 + 2p + 4 - 2p2 + 5 = 3p2 - 2p2 + 2p + 4 + 5 = p2 + 2p + 9

Back to Algebra index

Index laws

Multiplication

How can we work out 23 × 25?

23 = 2 × 2 × 2

25 = 2 × 2 × 2 × 2 × 2

so 23 × 25 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 28

There are 3 twos from 23 and 5 twos from 25, so altogether there are 8 twos.

In general, 2m × 2n =2(m + n)

Examples

25 × 24 = 2(5 + 4) = 29

27 × 23 = 2(7 + 3) = 210

The rule also works for other numbers, so

34 × 32 = 3(4 + 2) = 36

256 × 254 = 25(6 + 4) = 2510

Division

If you divide 25 by 23 you see that some of the 2's cancel:

So 25 ÷ 23 = 22

In general, 2m ÷ 2n = 2(m - n)

Example

25 ÷ 22 = 2(5 - 2) = 23

27 ÷ 23 = 2(7 - 3) = 24

The rule also works for other numbers, so

510 ÷ 53 =5(10 - 3) = 57

459 ÷ 454 = 45(9 - 4) = 455

Dividing numbers with roots or powers

Five 2s are divided by three 2s

More from Number

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