Rationalise the denominator of 10/(4 + √6), simplifying fully.
- Multiply top and bottom by (4 − √6).
- Bottom: (4 + √6)(4 − √6) = 42 − 6 = 16 − 6 = 10.
- Top: 10(4 − √6).
- Divide: 10(4 − √6)/10 = 4 − √6.
Answer: 4 − √6
Surds and exact calculation
Maths · Number. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
A surd is a root that does not work out exactly, such as √2 or √5. Its decimal goes on for ever, so the only exact way to write it is to leave the root sign. In the same way, an exact answer about a circle keeps π in it, for example 6π cm.
You simplify a surd by taking out the largest square number inside it. Surds behave like letters in algebra: you can add 3√2 and 5√2, but you cannot add √2 and √3 into one term.
To rationalise the denominator means to remove the surd from the bottom of a fraction without changing its value.
Rules to remember
- √a × √b = √(ab) and √a ÷ √b = √(a/b).
- √a × √a = a.
- Simplify by taking out the largest square factor: √48 = √16 × √3 = 4√3.
- Only like surds can be added: 3√2 + 5√2 = 8√2. √a + √b is not √(a + b).
- (a + √b)(a − √b) = a2 − b, with no surd left.
- To rationalise k/√a, multiply top and bottom by √a.
- To rationalise k/(a + √b), multiply top and bottom by (a − √b). Change the sign in the middle.
Common mistake
Adding the numbers under the roots, for example √9 + √16 = √25. In fact √9 + √16 = 3 + 4 = 7, and √25 = 5. Simplify each surd first, then add like surds only.
Worked example
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