Write 14/(3 − √2) in the form a + b√2.
- Multiply top and bottom by (3 + √2).
- Bottom: (3 − √2)(3 + √2) = 9 − 2 = 7
- Top: 14(3 + √2), so the fraction is 14(3 + √2)/7 = 2(3 + √2)
- 2(3 + √2) = 6 + 2√2
Answer: 6 + 2√2
Surds
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 to an exact fraction or whole number, such as √2 or √5. We leave it as a root so that the answer stays exact.
To simplify a surd, find the largest square number that divides the number under the root and take its root outside. For example √72 = √36 × √2 = 6√2. You can only add or subtract surds that have the same number under the root, just like collecting like terms.
To rationalise the denominator means to remove the surd from the bottom of a fraction. You multiply the top and the bottom by the same thing, so the value does not change.
Rules to remember
- √(ab) = √a × √b and √(a/b) = √a ÷ √b
- √a × √a = a
- Like surds: p√a + q√a = (p + q)√a, for example 3√2 + 5√2 = 8√2
- √a + √b is not equal to √(a + b)
- To rationalise k/√a, multiply top and bottom by √a: k/√a = k√a/a
- (a + √b)(a − √b) = a2 − b
- To rationalise k/(a + √b), multiply top and bottom by (a − √b); for k/(a − √b), use (a + √b)
Common mistake
Students add the numbers under the roots, for example writing √50 + √18 as √68. Simplify each surd first, then add only the like surds.
Worked example
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