IGCSE Maths / Working with algebra

Index laws with algebraic terms revision notes

Edexcel International GCSE Maths (4MA1), chapter 4: Working with algebra, specification point 2.1d-f. Free lesson.

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When algebraic terms have the same letter, handle the numbers and the powers separately. This lets you apply index laws correctly and avoid common errors with brackets.

Keep the numbers and powers separate

In a term such as 3a², 3 is the number in front and a² is the part with the letter. Work with these parts separately, then write them together. The numbers are multiplied or divided in the ordinary way. The letters follow the index laws. When multiplying powers with the same base, add the powers. When dividing powers with the same base, subtract the powers.

Multiply numbers and add powers

For 3a² × 4a³, multiply 3 and 4, then add the powers of a. This gives 12a⁵. You can also see why by writing out the letters: there are two a's and then three more, making five a's altogether.

Divide numbers and subtract powers

For 20b⁷ ÷ 5b², divide the numbers and subtract the powers of b. The result is 4b⁵. The two operations are separate, so do not apply the index rule to the numbers in front.

A bracket power applies to every factor

When a bracket is raised to a power, that power applies to everything inside it, including the number in front. In (2c³)², square the 2 and multiply the power of c by 2. This gives 4c⁶, not 2c⁶.

Avoid adding the numbers in front

The index rule for multiplication applies to the powers, not to the numbers in front. For example, 3 and 4 must be multiplied, not added. Adding them would give the incorrect answer 7a⁵ instead of 12a⁵. For 5x³ × 3x⁴, multiply 5 × 3 and add 3 + 4, giving 15x⁷. For (3y²)³, cube the 3 as well as multiplying 2 by 3, giving 27y⁶.