Examples · Basic Math
Square Root Examples: 10 Worked Problems
Ten roots, ordered so each adds one difficulty. The last three are where surds stop being arithmetic and start being algebra.
Perfect squares and simple simplifications
The first three square root examples need only the largest square factor.
- 12144 is a perfect square
- √(36 · 2)36 is the largest square factor
- 6√2√36 = 6
- √(100 · 2)100 is the largest square factor
- 10√2√100 = 10
Two and three simplify to multiples of the same surd, which is what makes them combinable later.
When the square factor is not obvious
Use prime factorisation and pair the primes. Each pair leaves the radical as one copy.
- 2 · 2 · 2 · 3 · 7 · 7the prime factorisation
- (2·2) and (7·7) pair offleaving 2 · 3 = 6 inside
- 14√62 × 7 = 14 comes out
- 2 · 7 · 7prime factors
- 7√2the pair of sevens escapes
This method never requires spotting a large square. It is slower than recognising 100 × 2, and it cannot fail.
Adding and multiplying surds
Adding radicals works like collecting like terms: only matching surds combine. Multiplying radicals is freer, because roots distribute over products — and the radicand of the answer is simply the product of the two radicands.
- 2√2 + 3√2simplify each first
- 5√2now they are like terms
- √3 + √5different surds
- no simpler formunlike terms do not combine
- √60roots multiply straight across
- 2√15then simplify: 60 = 4 · 15
Rationalising a denominator
Removing a root from the bottom of a fraction. Multiply top and bottom by that root.
- 3 ÷ √2a root on the bottom
- (3 · √2) ÷ (√2 · √2)multiply both parts by √2
- 3√2 ÷ 2the denominator is now a whole number
- (6√3) ÷ 3multiply top and bottom by √3
- 2√36/3 reduces to 2
The value has not changed in either case — multiplying by √2/√2 is multiplying by 1. Only the form has, and a whole-number denominator is easier to compare and to add.
Estimating before you start
A quick estimate catches a mistyped radicand before any work is wasted.
- 34² = 1156, 35² = 1225the perfect squares either side
- 34 < √1176 < 35so the answer is a little over 34
- 14√6 ≈ 34.29which matches the simplified form
Every root above is irrational except the first, so an exact surd is the only precise answer and the decimal is always an approximation. Estimating first tells you roughly what that decimal should be.
Checking the exact value against a decimal
Every simplification can be checked numerically in seconds.
| Example | Exact | Decimal | Original |
|---|---|---|---|
| 2 | 6√2 | 8.4853 | √72 = 8.4853 ✓ |
| 6 | 5√2 | 7.0711 | √8 + √18 = 7.0711 ✓ |
| 9 | 3√2/2 | 2.1213 | 3/√2 = 2.1213 ✓ |
If the decimals disagree, the simplification lost something. This catches the error that matters most: pulling out a factor that was not a perfect square.
Questions about square root examples
How do I simplify a square root?
Find the largest perfect square that divides the number, split the root into two, and take the root of the square part. √72 becomes √36 · √2, which is 6√2.
Can √3 + √5 be simplified?
No. Only matching surds combine, in the same way that only like terms combine in algebra. √3 + √5 is already in simplest form.
What does rationalising the denominator mean?
Removing a root from the bottom of a fraction by multiplying top and bottom by that root. 3/√2 becomes 3√2/2.
What if I cannot see the square factor?
Use prime factorisation. Break the number into primes and pair them — each pair leaves the radical as a single copy of that prime.