Combine after simplifying
Simplify √12 + √27.
- Extract square factors: √12 = 2√3 and √27 = 3√3.
- Both terms now contain √3, so add their coefficients.
Answer: 5√3
Surds and rationalising
Simplify before combining, and rationalise without changing the value.
Before you start
You’ll need Fractional and negative indices. Follow a link if you want to revise it first.
Simplifying surds
A square factor can leave a radical: √(a²b) = |a|√b. Use the absolute value because a may be negative.
Only like radicals combine. Simplify each radical first, then add the coefficients of terms with the same radical.
Rationalising means multiplying numerator and denominator by the same nonzero expression. The resulting fraction has the same value.
For example, √2 × √3 = √6, whereas √2 + √3 must stay as two terms.
Adding surds and rationalising a fraction
Use a conjugate
Rationalise 1/(2 + √3).
- Multiply top and bottom by 2 − √3.
- The denominator becomes 4 − 3 = 1.
Answer: 2 − √3
Combining radicands during addition changes the value. Extract square factors first.
Practice questions
The first link uses smaller values. Move on when you can answer without referring to the examples. The difficulty settings aren’t tied to exam grades.
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Check your understanding
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Why is √(x²) equal to |x| rather than always x?