Index laws

Multiplying and dividing

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

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

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

Example

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