To simplify a root, find a square number that divides the number under the root, take its square root outside, and leave the rest inside. The result is exact, unlike a calculator decimal. You will use it in Pythagoras questions, in algebra, and whenever a question says “leave your answer in surd form”.
This lesson needs the square numbers and a little fluency with factors and exact arithmetic. It pairs with the index law a1/2 = √a from applying index laws.
How does the method work?
The key fact is that √(a × b) = √a × √b. So if the number under the root contains a square factor, that part of the root comes out as a whole number.
- List the square numbers: 4, 9, 16, 25, 36, 49, 64, 81, 100.
- Find the largest one that divides the number under the root.
- Split the number into that square times the remaining factor.
- Take the square root of the square factor and write it in front.
- Check by squaring the whole answer, or by multiplying the coefficient back in.
Worked example
Simplify √75 + √12, giving your answer in the form k√3.
Step 1, simplify √75. The largest square factor of 75 is 25, because 75 = 25 × 3. So √75 = √25 × √3 = 5√3.
Step 2, simplify √12. The largest square factor of 12 is 4, because 12 = 4 × 3. So √12 = √4 × √3 = 2√3.
Step 3, add like terms. 5√3 + 2√3 = 7√3, in the same way that 5 apples + 2 apples = 7 apples.
Step 4, check. 7√3 squared is 49 × 3 = 147. And (√75 + √12)² = 75 + 12 + 2√900 = 87 + 60 = 147. Both give 147.
Answer: 7√3
The mistake to watch for
The usual slip is to write the square factor itself outside the root, instead of its square root.
Mistaken working: √72 = √(36 × 2) = 36√2
The student took 36 out of the root without square-rooting it.
The correction: when a factor leaves the root, it is square-rooted on the way out. √36 = 6, so √72 = 6√2.
A quick size check exposes the slip. √72 is a little above 8, because 8² = 64 and 9² = 81. And 6√2 ≈ 6 × 1.41 = 8.5, which fits, while 36√2 is about 51, which does not.
Check yourself
Try these without a calculator, then open each answer.
1. Simplify √45.
Show answer
45 = 9 × 5, so √45 = √9 × √5 = 3√5.
2. Simplify √98.
Show answer
98 = 49 × 2, so √98 = 7√2. Answer: 7√2.
3. Simplify √12 × √6.
Show answer
√12 × √6 = √72, since 12 × 6 = 72. From the worked method, √72 = 6√2.
Where this leads next
The next lesson, estimating the size of a standard-form answer, uses the same habit of checking whether a result is sensible. After that, try the indices, roots and standard form practice set. Later in the subject, roots return in right-angled triangle work, so right-triangle trigonometry builds on this skill.
If you can simplify a root when prompted but not when it is hidden inside a longer question, that gap is something online one-to-one Mathematics tuition can target directly.