Simplifying surds. How to write root 12 in the form a root b.

Sometimes you will be asked to simplify a surd or write the surd in the form a√b. 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 √(a×b) = √a√b to give the surd in the form a√b.

Example 1

Write √24 in the form a√6.

The question is asking you to simplify the surd and write it in the form a√b. 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:

√24 = √(4×6)

Next use the rule √(a×b) = √a√b

√(4×6) = √4√6

Finally square root the 4 to give 2:

√4√6 = 2√6

Example 2

Write √18 in the form a√2.

The question is asking you to simplify the surd and write it in the form a√b. 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:

√18 = √(9×2)

Next use the rule √(a×b) = √a√b

√(9×2) = √9√2

Finally square root the 9 to give 3:

√9√2 = 3√2

Example 3

Write √75 in the form a√3.

The question is asking you to simplify the surd and write it in the form a√b. 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:

√75 = √(25×3)

Next use the rule √(a×b) = √a√b

√(25×3) = √25√3

Finally square root the 25 to give 5:

√25√3 = 5√3

Example 4

Write √12 in the form a√b.

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:

√12 = √(4×3)

Next use the rule √(a×b) = √a√b

√(4×3) = √4√3

Finally square root the 4 to give 2:

√4√3 = 2√3

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