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Index notation

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


What is 3c2 × 5c4?

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To work it out:

  • Add the indices:
  • 2 + 4 = 6
  • Multiply the numbers in front of the variable:
  • 3 x 5 = 15
  • Answer:
  • 3c2 × 5c4 = 15c6

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


What is s2 + 8s2 - 2s2?

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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.


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

Back to Algebra index

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