Numbers with square roots also follow fixed rules, which let them be tidied into simpler forms.
Factorising the number under the root, as in √12 = √(2 × 2 × 3), shows a pair of 2s, and that pair comes out of the root as a single 2, giving 2√3.
Terms with the same root, such as 2√3 and 5√3, combine the way like terms do in algebra, by adding their coefficients, to give 7√3, while multiplication and division work on the numbers under the roots.
If a root is left in a denominator, multiplying top and bottom by the same root makes the denominator rational; this is called rationalising the denominator.
Simplifying by factorising under the root
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Factorising the number n under the root and taking paired factors outside
12
🔧 Why simplifying works
①A product of two equal numbers a² comes out of the root as a: √(a² × b) = a√b
②√12 = √(2² × 3) = 2√3.
③√48 = √(4² × 3) = 4√3.
④If only unpaired prime factors remain, it cannot be simplified further; in the picture, n = 30 is 2 × 3 × 5 with no pairs at all.
Add and subtract only like roots
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Two bars measured in units of length √2 joined end to end, so that the coefficients add
3
2
📏 The key to addition
①Coefficients can be added or subtracted only when the roots are the same, just as with like terms in algebra.
②2√3 + 5√3 = 7√3.
③2√3 + 5√2 cannot be simplified, because the roots differ.
④Roots that look different sometimes become the same after simplifying: √12 + √27 = 2√3 + 3√3 = 5√3
Multiplying and dividing roots
Multiplication
√a × √b = √(ab) (a > 0, b > 0)
When roots are multiplied, the numbers under them are multiplied.
Division
√a√b = √(ab)
When roots are divided, the numbers under them are divided.
💡 Care points
①√2 × √3 = √6.
②√6 ÷ √2 = √3.
③(2√3)² = 2² × (√3)² = 4 × 3 = 12: the coefficient is squared too.
④Unlike multiplication, addition cannot combine what is under the roots: √a + √b ≠ √(a + b)
Rationalising the denominator
Rationalising the denominator
a√b = a√bb
Multiply top and bottom by √b to remove the root from the denominator.
🧹 Why rationalise
①With a root in the denominator, values are hard to compare or estimate.
②Multiplying top and bottom by the same root makes the denominator rational, because (√b)² = b.
③For example, 1/√2 = √2/2 ≈ 0.707.
Worked examples
Example 1
Compute √2 × √8.
1
When roots are multiplied, multiply the numbers under them.
√2 × √8 = √(2 × 8) = √16
2
16 is the square of 4, so √16 = 4.
= 4
▸ 4
If the product under the root is a perfect square, the root disappears.
Example 2
Compute 2√3 + 5√3.
1
The roots are the same, so add the coefficients.
(2 + 5)√3
2
2 + 5 = 7.
= 7√3
▸ 7√3
Roots must be the same before they can be added, just as with like terms.
Exam checklist
Working with roots
a√m ± b√m = (a ± b)√m
Add or subtract only the coefficients of terms with the same root.
Grade-9 school exam type
Rationalize the denominator of 6/√3.
①2√3
②3√3
③√3
④6√3
⑤2
▸ ① 2√3
1
Multiply numerator and denominator by √3.
6/√3 × √3/√3 = 6√3/3
2
Reduce.
= 2√3
🎯 Exam points
①Simplify first: √48 + √12 = 4√3 + 2√3 = 6√3.
②Rationalise so that no root is left in the denominator of the answer.
③Use (√a)² = a.
④√a × √b = √(ab), but √a + √b ≠ √(a + b).
⑤A negative number cannot go under a square root (among real numbers).