Method · Basic Math
How to Simplify a Square Root Step by Step
√72 and 6√2 are the same number. The second is the finished answer, and getting there is a search for one thing: the largest square factor.
What simplifying actually means here
Learning how to simplify a square root is learning to split the number under the radical into two parts: a perfect square that can come out, and whatever is left behind. √72 becomes 6√2 because 72 is 36 × 2, and 36 is a perfect square whose root is a whole number.
The result is not an approximation. 6√2 is exactly √72 — the same value written so that nothing further can be extracted. A decimal like 8.485 is the approximation, and it is the form that loses information.
The method, in three steps
Find the largest perfect square that divides the number, split the root into two, then take the root of the square part.
- √72the expression
- √(36 · 2)36 is the largest perfect square dividing 72
- √36 · √2a root of a product is the product of the roots
- 6√2√36 = 6, and 2 has no square factor left
The middle step is the one that makes the method legal: √(ab) = √a · √b. That identity holds for multiplication and for division. It does not hold for addition, which is the source of the worst mistake in this topic.
When you cannot spot the square factor
For awkward numbers, use prime factorisation instead of guessing. Break the number into primes, then pair them up — every pair escapes the root as a single copy.
- 200 = 2 × 2 × 2 × 5 × 5
- Pair them: (2 × 2) and (5 × 5), with one 2 left over
- Each pair leaves as a single factor: 2 × 5 = 10, and the leftover 2 stays inside
- So √200 = 10√2
This always works and never requires recognising a large square on sight. It is slower than spotting 100 × 2 directly, but it cannot fail.
The mistakes that cost marks
Three errors account for nearly all lost marks on this topic, and each has a quick check that catches it.
| The mistake | Why it is wrong | The check |
|---|---|---|
| √(9 + 16) = 3 + 4 | roots do not distribute over addition | √25 = 5, not 7 |
| √72 = 2√18 | 18 still contains the square 9 | is anything left inside square-free? |
| √2 ≈ 1.41 early on | rounding compounds through later steps | keep the root exact until the end |
The first is worth dwelling on. Square roots distribute over products and quotients, never over sums or differences. There is no way to simplify √(a + b) into separate roots, and any method that appears to is wrong.
Which roots are irrational
The square root of a whole number is either a whole number or irrational — there is no middle case. √49 is exactly 7. √50 cannot be written as any fraction at all, so the exact form 5√2 is the only precise way to write it.
This is why simplified radical form matters in the first place. It is not decoration; for most numbers it is the only exact answer available.
The vocabulary, so the instructions make sense
Textbooks use precise names for the parts of a radical, and instructions become much clearer once you have them.
- The radical is the √ symbol itself. To simplify a radical is to do exactly what this page describes.
- The radicand is the number underneath it. In √72 the radicand is 72, and the whole method is a search for square factors of the radicand.
- A surd is an irrational root left in exact form. 6√2 is a surd; 8.485 is its decimal approximation.
So 'write √72 in simplest surd form' and 'simplify the square root of 72' are the same instruction, and the exact value in both cases is 6√2.
Using prime factorization on awkward numbers
Prime factorization is the fallback when no square factor is obvious. Break the radicand into primes, pair them, and each pair leaves the root as one copy of itself.
For √1176: the primes are 2 × 2 × 2 × 3 × 7 × 7. The pairs are (2 × 2) and (7 × 7), leaving 2 × 3 = 6 inside. So the exact value is 2 × 7 × √6, which is 14√6. No guessing was needed at any point.
Rationalising a denominator
A related instruction you will meet is to rationalise the denominator — to remove a root from the bottom of a fraction. Multiply the top and bottom by that root.
For 3/√2, multiply both parts by √2 to get 3√2/2. The value has not changed, but the denominator is now a whole number, which makes the fraction easier to compare and to add to others.
Roots of negative numbers
There is no real number whose square is negative, so √−4 has no value among the real numbers. It becomes possible once imaginary numbers are introduced, where i is defined as √−1 and √−4 is 2i. Until then, a negative under a square root means the question has no real answer.
Questions about how to simplify a square root
How do I know when a square root is fully simplified?
When the number left under the radical has no perfect square factor above 1. Check by testing 4, 9, 25 and 49 — if none of them divides it, you are finished.
What is √72 in simplest form?
6√2. The largest perfect square dividing 72 is 36, whose root is 6, and the remaining 2 has no square factor. As a decimal it is about 8.4853, but 6√2 is exact.
Can you simplify √(a + b)?
No. Square roots distribute over multiplication and division only. √(9 + 16) is √25, which is 5 — not 3 + 4.
Why keep the root instead of using the decimal?
Because the decimal is rounded and the root is exact. If the value feeds into another calculation, early rounding compounds, and exam answers are usually expected in exact radical form anyway.