Page 4 - Lesson Notes

2. Multiply BothNumerator and denominator by the Conjugate or Rationalising
factor of the denominator.

There is another special way to move a square root from the bottom of a fraction to
the top ... we multiply both Numerator and denominatorby the conjugate of the
denominator.

The conjugate is where we change the sign in the middle of two terms.

Conjugate of a+√b is a –√b and Conjugate of √a –√b is √a+√b

Examples: Conjugate of 2+√3 is 2-√3.

Conjugate of√5-√6 is √5+√6

Because when we multiply something by its conjugate we get squares like this:

2
2
(a+b)(a−b) = a − b
Example: Here is a fraction with an "irrational denominator":

1/(3−√2)

How can we move the square root of 2 to the top?

We can multiply both top and bottom by 3+√2 (the conjugate of 3−√2), which
won't change the value of the fraction:

2
2
1(3+√2) /(3−√2) ×( 3+√2) = (3+√2)/3 −(√2) = (3+√2)/7
2
2
(we used (a+b)(a−b) = a − b in the denominator)

Surds

 If a is a rational number and n is a positive integer such that the nth root of a
is an irrational number, then a 1/n is called a surd.

 If a is a surd then ‘n’ is known as order of surd and ‘a’ is known as
radicand.
 Every surd is an irrational number but every irrational number is not a sure.
(For example: 2 + 1 is irrational but not a surd)

 Laws of Radical:

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