A surd keeps an irrational root in exact form. The square root of 11 cannot be replaced by a terminating decimal without losing precision, so √11 is often the most useful answer. These surds worked examples show how to simplify roots, collect like terms and rationalise a denominator while preserving the original value.
The key habit is to simplify before deciding what can be combined. Two terms that initially look different may contain the same irrational part. On the other hand, two expressions containing square-root signs are not automatically like terms.
Start with a square-factor search
For nonnegative a and b, √(ab) = √a × √b. This means a perfect-square factor can come outside a square root. For example, √63 = √(9 × 7) = 3√7. The number 3 is the coefficient; √7 is the remaining surd part.
Useful square factors to test are 4, 9, 16, 25, 36, 49 and 64. Try a large factor first, but do not worry if you miss the largest one. Repeated correct simplification reaches the same answer. A root is fully simplified when its integer radicand, the number beneath the root sign, has no square factor greater than one.
In this lesson, all numerical square roots mean their nonnegative principal values. Thus √36 = 6. The equation x2 = 36 has two solutions, 6 and −6, but that is a different question.
Worked example 1: expose the like surds
Question: Simplify 2√75 − √48 + √27.
- Since 75 = 25 × 3, √75 = 5√3. The first term is therefore 2 × 5√3 = 10√3.
- Since 48 = 16 × 3, √48 = 4√3.
- Since 27 = 9 × 3, √27 = 3√3.
- Collect the coefficients: 10√3 − 4√3 + 3√3 = 9√3.
The irrational part stays unchanged, just as 10x − 4x + 3x = 9x. An answer of √54 obtained by combining the numbers inside the roots would be incorrect. Addition and subtraction do not follow the multiplication rule for roots.
For a rough check, √3 is about 1.73, so the answer is about 15.6. Evaluating the original three terms gives approximately the same value. The decimal is a check, not a replacement for the exact result.
Decide what kind of operation you have
| Operation | Useful rule | Example |
|---|---|---|
| Addition | Combine identical surd parts | 3√5 + 2√5 = 5√5 |
| Multiplication | Multiply coefficients and radicands | 2√3 × 4√6 = 8√18 = 24√2 |
| Squaring | Square the entire expression | (3√2)2 = 9 × 2 = 18 |
Terms such as √2 + √7 cannot be combined into one like-surd term. Leaving a sum of different simplified roots is a valid exact answer. A question asking you to simplify does not mean every plus sign must disappear.
Worked example 2: a denominator with one surd term
Question: Rationalise 9/(2√3).
A rational denominator contains no irrational root. Multiply the numerator and denominator by √3. The multiplier √3/√3 is one, so this changes the form but not the value:
9/(2√3) = (9√3)/(2√3 × √3) = (9√3)/6 = (3√3)/2.
There are two separate operations here: remove the root from the denominator, then cancel a common numerical factor. The 9 and 6 have a common factor of 3. Do not cancel the 3 beneath a root against an unrelated 3 outside a root; those are not matching factors.
To check, multiply the final answer by the original denominator: ((3√3)/2) × 2√3 = 3 × 3 = 9. This recovers the original numerator and verifies the rationalisation without any rounding.
Worked example 3: a conjugate removes the middle terms
Question: Rationalise 5/(√7 − 2).
The conjugate of √7 − 2 is √7 + 2. It changes the sign between the two terms. Multiply both parts of the fraction by that conjugate:
5/(√7 − 2) = 5(√7 + 2)/((√7 − 2)(√7 + 2)).
Expand the denominator before simplifying:
7 + 2√7 − 2√7 − 4 = 3.
The two middle terms cancel, leaving a rational number. The result is (5√7 + 10)/3.
Multiplying only by √7 would not work as neatly: the denominator would become 7 − 2√7, which is still irrational. The conjugate is chosen because the difference-of-squares pattern removes both middle terms.
As a size check, √7 is approximately 2.646. The original denominator is positive but smaller than one, so dividing 5 by it should give an answer larger than 5. The final expression is approximately 7.743, consistent with that expectation.
Worked example 4: expand before you collect
Question: Simplify (√6 + √2)2.
Write the square as a product of two brackets. Multiplying every term gives 6 + √12 + √12 + 2. Since √12 = 2√3, the result is 8 + 4√3.
The common incorrect answer, 8, omits the two cross-products. Squaring a sum is different from adding two squares. To keep track, write (a + b)2 as a2 + 2ab + b2, then substitute the actual terms.
This example also explains why √(a + b) cannot generally be replaced by √a + √b. If those expressions were always equal, their squares would match, but the right-hand square normally contains an extra cross-term.
Three exact-answer practice questions
Cover the solutions and work each question on paper. Leave roots in exact form. At the end, check that every remaining square root is simplified and every denominator is rational.
Practice 1: simplify and subtract
Simplify √98 − 2√8 + √50.
Solution: The terms become 7√2 − 4√2 + 5√2. Combining their coefficients gives 8√2. Remember that the coefficient 2 in 2√8 multiplies the simplified value 2√2, giving 4√2.
Practice 2: choose the correct conjugate
Rationalise 4/(√6 + √2).
Solution: Multiply by (√6 − √2)/(√6 − √2). The denominator becomes 6 − 2 = 4, so the fraction is 4(√6 − √2)/4 = √6 − √2. Multiplying this by the original denominator gives 6 − 2 = 4, confirming the numerator.
Practice 3: multiply two surd brackets
Simplify (2√5 − √3)(√5 + 2√3).
Solution: The four products are 10, 4√15, −√15 and −6. Combining them gives 4 + 3√15. The negative sign belongs to both products arising from −√3; losing the last minus would change the constant term.
Specific questions about surds
Is every square root a surd?
No. A root that simplifies to a rational number is not a surd. For example, √49 = 7, whereas √13 is irrational and remains a surd.
Can I simplify √12 as √4 + √8?
No. A square root does not distribute over addition. Factor 12 as 4 × 3 instead, giving √12 = 2√3.
How do I know that I have finished simplifying?
Check for a remaining square factor in each integer radicand, combine all like surds and reduce any ordinary fraction. Rationalise the denominator if required.
Why multiply the numerator as well as the denominator?
Multiplying both by the same nonzero expression preserves the fraction’s value. Changing only the denominator creates a different number.
What is the conjugate of 3 + 2√5?
It is 3 − 2√5. Their product is 32 − (2√5)2 = 9 − 20 = −11, which is rational.
Does a negative rationalised denominator mean the method failed?
No. A negative denominator is still rational. You can move its minus sign to the numerator and simplify the expression.
Should I enter the root into a calculator first?
Use a decimal evaluation only for checking or when an approximate answer is requested. Early rounding can hide exact cancellation and introduce avoidable error.
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