Number Systems
Surds and real numbers
Simplify, add and multiply surds, use fractional powers, and clear a surd out of a denominator.
A surd is a root that refuses to come out whole — √2, √3, √10. You cannot write it as a neat decimal, so you leave it as it is and work with it directly. That sounds harder than it is: surds add like letters in algebra, and multiply by simply going under the same root sign.
The two rules that do most of the work
√a × √b = √(ab), so √3 × √12 = √36 = 6. And a√c + b√c = (a + b)√c, so 2√5 + 3√5 = 5√5. What you cannot do is add across different roots: √2 + √3 stays exactly as it is.
Worked example
Simplify √72.
Look for the largest perfect square that divides 72. It is 36, since 72 = 36 × 2.
Pulling out the biggest square in one go saves simplifying twice over.
Try it together
Work out (2 + √3)(2 - √3).
Two brackets that differ only by a sign are a difference of two squares, and the surd disappears.
1.First square the front term: what is 2 × 2?
A fractional power says root then power
The bottom of the fraction is the root and the top is the power: 8 to the power 2/3 means take the cube root of 8, giving 2, then square it, giving 4. Doing it in that order keeps the numbers small.
Have a go
Have a go on your own: √48 can be written as a√3. What is a?
Hint: Which perfect square divides 48?
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.