Number
Grade 6-8Surds
A surd is a root that cannot be simplified to a whole number, and keeping calculations in surd form gives an exact answer instead of a rounded decimal. This lesson covers simplifying surds, adding and subtracting like surds, multiplying surds and expanding brackets, and rationalising the denominator.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Simplifying surds
To simplify a surd, find the largest square number that divides into the number under the root, and split it into two roots multiplied together. The square root of the square number simplifies to a whole number, leaving a smaller surd.
√50: the largest square factor of 50 is 25, so √50 = √25 × √2 = 5√2.
Adding and subtracting surds
Surds can only be added or subtracted directly if the number under the root is the same, in the same way that only like terms can be collected in algebra. If the surds look different, simplify them first to see if they become like surds.
√8 + 3√2: simplify √8 to 2√2 first, so the expression becomes 2√2 + 3√2 = 5√2.
Multiplying surds and expanding brackets
To multiply two surds, multiply the numbers under the roots together and take the root of the result. When a surd multiplies itself, the surd disappears entirely, since √a × √a = a.
√3 × √12 = √36 = 6. Expanding √2(3 + √2) = 3√2 + √2 × √2 = 3√2 + 2.
Rationalising the denominator
A fraction with a surd on its own in the denominator is rationalised by multiplying the top and bottom by that surd. A fraction with a sum or difference involving a surd in the denominator, like 2 + √3, is rationalised by multiplying the top and bottom by the conjugate, which flips the sign in the middle, since multiplying a bracket by its conjugate removes the surd using the difference of two squares.
5/√5 = (5 × √5)/(√5 × √5) = 5√5/5 = √5. For 1/(2 + √3), multiply by (2 - √3)/(2 - √3): the denominator becomes (2 + √3)(2 - √3) = 4 - 3 = 1, giving 2 - √3.
Worked Examples
Three exam-style questions, fully solved.
Simplify √50.
Easy- 1.Find the largest square factor of 50: 25
- 2.Split into two roots: √25 × √2
- 3.Simplify √25 to 5
Answer: 5√2
Rationalise the denominator of 5/√5. Give your answer in its simplest form.
Medium- 1.Multiply the top and bottom by √5: (5 × √5) / (√5 × √5)
- 2.Simplify the denominator: √5 × √5 = 5
- 3.Simplify the fraction: 5√5 / 5
Answer: √5
Rationalise the denominator of 1 / (2 + √3).
Hard- 1.Multiply the top and bottom by the conjugate, 2 - √3
- 2.Simplify the denominator using the difference of two squares: (2 + √3)(2 - √3) = 4 - 3 = 1
- 3.Simplify the numerator: 1 × (2 - √3) = 2 - √3
Answer: 2 - √3
Avoid These
The most common mistakes students make.
Not using the largest square factor when simplifying a surd, leaving the answer only partially simplified.
Adding or subtracting surds with different numbers under the root as if they were like terms, instead of simplifying first to check whether they match.
Forgetting that √a × √a = a when expanding brackets or multiplying surds, and leaving the result as a surd instead of a whole number.
Rationalising a denominator with a single surd by multiplying only the denominator, and forgetting to multiply the numerator by the same surd.
Multiplying by the same expression instead of the conjugate when rationalising a denominator with a sum or difference of a surd, which leaves a surd in the denominator instead of removing it.
FAQ
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