Remove a surd from the bottom of a fraction, with √k or a bracket like (b + √k) underneath.
What you will learn
- Rationalise a/√k and a/(b√k)
- Rationalise a/(b + √k) with the conjugate
- Write answers in the form p + q√k
A single surd underneath
A surd on the bottom of a fraction is an irrational denominator.
To rationalise the denominator means to rewrite the fraction with no surd on the bottom. Multiply the top and the bottom by the surd. The value does not change (you multiply by 1), and √k × √k = k.
- 6/√3
- = 6√3/3
- = 2√3
With a number in front, multiply by the surd only: 10/(4√5) = 10√5/(4 × 5) = 10√5/20 = √5/2.
A sum on top splits into two fractions: (6 + √2)/√2 = 6/√2 + √2/√2 = 3√2 + 1.
To add fractions like 1/√2 + 3/√8, rationalise each one: 1/√2 = √2/2 and 3/√8 = 3/(2√2) = 3√2/4, so the total is 2√2/4 + 3√2/4 = 5√2/4.
The rest of this lesson and its 7 practice questions are locked.
In this lesson
- A single surd underneath
- A bracket underneath
7 practice questions, marked as you go; about 25 minutes in all.
Key terms
Rationalise the denominator, Conjugate
In this unit: Surds
Exact answers with square roots: simplifying, multiplying and expanding, rationalising denominators, and surds in Pythagoras and quadratics. Higher tier.
- Simplifying surds
- Multiplying and expanding surds
- Rationalising the denominator
- Exact answers with surds