Sometimes you will be asked to simplify a surd or write the surd in the form avb. To do this you will need to think of two integers that multiply together to give the number underneath the surd. One of the numbers must be a square number so that you can take the square root of this number. All you need to do now is use the rule v(a?b) = vavb to give the surd in the form avb.

Example 1

Write v24 in the form av6.

The question is asking you to simplify the surd and write it in the form avb. The value of b is already found (b=6)

Two integers that multiply to give 24 are 4 and 6 (and 4 is a square number), so let’s write 24 as 4 ? 6:

v24 = v(4?6)

Next use the rule v(a?b) = vavb

v(4?6) = v4v6

Finally square root the 4 to give 2:

v4v6 = 2v6

Example 2

Write v18 in the form av2.

The question is asking you to simplify the surd and write it in the form avb. The value of b is already found (b=2)

Two integers that multiply to give 18 are 9 ? 2 (and 9 is a square number), so let’s write 18 as 9 ? 2:

v18 = v(9?2)

Next use the rule v(a?b) = vavb

v(9?2) = v9v2

Finally square root the 9 to give 3:

v9v2 = 3v2

Example 3

Write v75 in the form av3.

The question is asking you to simplify the surd and write it in the form avb. The value of b is already found (b=3)

Two integers that multiply to give 75 are 25 and 3 (and 25 is a square number), so let’s write 25 as 25 ? 3:

v75 = v(25?3)

Next use the rule v(a?b) = vavb

v(25?3) = v25v3

Finally square root the 25 to give 5:

v25v3 = 5v3

Example 4

Write v12 in the form avb.

Two integers that multiply to give 12 are 4 and 3 (and 4 is a square number), so let’s write 12 as 4 ? 3:

v12 = v(4?3)

Next use the rule v(a?b) = vavb

v(4?3) = v4v3

Finally square root the 4 to give 2:

v4v3 = 2v3

Related video: Write surds in the form – GCSE maths level 7 question