The classic: root 72
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sqrt(72)72 = 2 cubed x 3 squared. The pairs are 2x2 and 3x3, which send a 2 and a 3 outside, leaving one 2 inside. So root 72 = 6 root 2, and squaring it back gives 36 x 2 = 72.
Radical Calculator
Simplify, combine and rationalize radicals - the answer is a form, not a decimal.
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The answer to a radical question is a form, not a decimal
A calculator will tell you that the square root of 72 is 8.4852814. That is almost never the answer a radical question wants. What it wants is 6 times the square root of 2 - the same quantity written so that nothing perfect is left hiding under the sign. That difference is the whole subject: simplifying a radical is a rewrite, not an evaluation, and a decimal throws away the exactness you were being asked to keep.
Everything on this page is exact. The square root of 72 is 6 root 2, the cube root of 54 is 3 cube-root 2, the square root of two thirds is root 6 over 3, and one over two plus root 3 is exactly 2 minus root 3. A decimal is offered beside each answer, clearly labelled as an approximation, because sometimes you do want to know roughly how big the thing is - but it is never handed to you instead of the answer.
Three questions share this page because they share one skill. Simplifying finds the largest perfect power inside the radicand. Adding tells you whether two radicals are like terms, which you cannot know until both are simplified. Rationalizing moves a root out of a denominator using either the root itself or the conjugate. All three come down to the same rule: a root of a product is the product of the roots.
What to notice
- Take out the LARGEST perfect square, not the first one you spot. 72 is 4 times 18, and stopping there leaves root 18 - still not simplest form. It is also 36 times 2, and 6 root 2 is finished.
- The prime factorization is where the perfect power lives. Once 72 is written as 2 cubed times 3 squared, every pair of equal primes is one factor that can step outside the square-root sign.
- Two radicals add only when the numbers left under their signs are identical. Root 8 plus root 18 works, because both are secretly multiples of root 2. Root 8 plus root 27 does not, and 2 root 2 plus 3 root 3 is the finished answer.
- A fraction under the sign, and a root in a denominator, are the same problem. Root of two thirds is not simplest form; multiplying inside by 3 over 3 turns it into root 6 over 3, which is.
- The index says which power comes out. Under a square root you need pairs, under a cube root triples, under a fourth root groups of four. That is why the cube root of 54 gives 3, not 6.
How to simplify a radical by hand
Factor the radicand into primes
Write the number under the sign as a product of primes. For 72 that is 2 x 2 x 2 x 3 x 3, or 2 cubed times 3 squared. Doing this first means you never have to guess whether a perfect square is hiding in there.
Group the primes by the index
For a square root, gather the primes into pairs; for a cube root, into triples. In 72 the pairs are 2 x 2 and 3 x 3, and one 2 is left over on its own.
One factor leaves the sign for each complete group
Each pair contributes a single copy of that prime outside the radical. The pair of 2s gives a 2 and the pair of 3s gives a 3, so 6 comes out, and the lonely 2 stays underneath: 6 root 2.
Clear any fraction from under the sign
If a denominator is left inside, multiply the inside top and bottom by whatever makes the bottom a perfect power. For two thirds, multiply by 3 over 3 to get 6 over 9, and root 6 over 3 comes out.
Check by squaring your answer back
6 root 2, squared, is 36 times 2, which is 72. That check catches the most common error in one line - taking out too small a factor leaves a square that is short of the radicand.
Radicals in simplest form
The cases that come up most, with the perfect power that does the work named in each one.
| Radical | Perfect power inside | Simplest form | Approximately |
|---|---|---|---|
| √8 | 4 | 2√2 | 2.828 |
| √12 | 4 | 2√3 | 3.464 |
| √18 | 9 | 3√2 | 4.243 |
| √32 | 16 | 4√2 | 5.657 |
| √50 | 25 | 5√2 | 7.071 |
| √72 | 36 | 6√2 | 8.485 |
| √(2/3) | none - clear the 3 | √6 / 3 | 0.816 |
| ∛54 | 27 | 3∛2 | 3.780 |
| 3 / √2 | none - rationalize | 3√2 / 2 | 2.121 |
| 1 / (2 + √3) | none - conjugate | 2 - √3 | 0.268 |
Worked examples
A coefficient in front: 3 root 8
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3*sqrt(8)Simplify the radical first: root 8 is 2 root 2. Then multiply the coefficients: 3 x 2 = 6, so the answer is 6 root 2. The coefficient never goes under the sign.
Adding what looks unaddable: root 8 + root 18
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sqrt(8) + sqrt(18)Neither term mentions root 2 until you simplify: root 8 is 2 root 2 and root 18 is 3 root 2. Now they are like terms, so add the coefficients: 5 root 2.
Unlike radicals: root 8 + root 27
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sqrt(8) + sqrt(27)Simplified, these are 2 root 2 and 3 root 3. Different numbers under the signs means they are not like terms, so 2 root 2 + 3 root 3 is the answer - there is nothing left to do.
A root in the denominator: 3 over root 2
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3/sqrt(2)Multiply top and bottom by root 2. The value does not change, because root 2 over root 2 is 1, and the bottom becomes 2. The answer is 3 root 2 over 2.
The conjugate: 1 over 2 plus root 3
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1/(2 + sqrt(3))Multiply top and bottom by 2 minus root 3. The bottom is then 2 squared minus root 3 squared, which is 4 - 3 = 1, so the whole thing is exactly 2 - root 3.
Common mistakes
- Taking out the first perfect square rather than the largest. Writing root 72 as 2 root 18 is true but unfinished, and it is marked wrong as an answer to "simplest radical form".
- Answering with a decimal. 8.485 is not the square root of 72; it is a rounding of it. If the question uses a radical sign, the answer keeps one unless the root comes out whole.
- Adding the numbers under the signs. Root 4 plus root 9 is 2 + 3 = 5, not root 13. Roots add only as like terms, never inside the sign.
- Multiplying radicals of different indexes. Root 2 times cube-root 2 is not root 2 or cube-root 4 - the product rule needs both indexes to match before you may combine them.
- Leaving a root in a denominator. Root 2 over 2 and 1 over root 2 are the same number, but only the first is in simplest form; the second still has a root downstairs.
- Treating root of a sum as a sum of roots. Root of 9 + 16 is root 25 = 5, not 3 + 4 = 7. The product rule splits products, not sums.
- Forgetting that an even root of a negative number is not real, while an odd one is. There is no real root 4 of -16, but the cube root of -8 is exactly -2.
Radical FAQ
What is simplest radical form?
A radical is in simplest form when the number under the sign has no perfect-power factor left in it, no fraction inside the sign, and no radical in a denominator. So 6 root 2 is simplest form and 2 root 18 is not, even though they are the same number.
How do I simplify a square root?
Factor the radicand into primes, gather the primes into pairs, and send one copy of each pair outside the sign. Anything unpaired stays underneath. Root 72 becomes 6 root 2 because 72 is 2 cubed times 3 squared.
Why not just give the decimal?
Because a decimal is an approximation and a radical is exact. Root 2 has no finite decimal, so 1.414 is already wrong in the fourth place; exam answers, further algebra and any later cancellation all need the exact form to stay correct.
When can two radicals be added?
Only when the numbers left under their signs are the same after simplifying, and the indexes match. Then they behave exactly like x + x: add the coefficients and keep the radical. Root 8 plus root 18 is 5 root 2; root 8 plus root 27 cannot be combined at all.
How do I rationalize a denominator?
If the denominator is a single root, multiply the top and bottom by that root. If it is a sum or difference like 2 plus root 3, multiply by its conjugate, 2 minus root 3 - the difference of squares that follows has no root in it. Either way you are multiplying by 1, so the value never changes.
What is a conjugate?
The same two terms with the sign between them flipped: the conjugate of 3 minus 2 root 2 is 3 plus 2 root 2. Multiplying a binomial by its conjugate gives a difference of squares, and squaring a square root removes it - which is what clears the denominator.
Can I simplify cube roots and fourth roots here?
Yes, any index from 2 to 6. The rule generalises: under a cube root you look for triples of equal primes rather than pairs, so the cube root of 54 is 3 cube-root 2 because 54 is 27 times 2.
Why does √(2/3) not equal √2/√3?
It does - they are the same number, and neither is in simplest form. The square root of the fraction 2/3 equals √2 divided by √3, and multiplying top and bottom by √3 gives √6 over 3, which has no root in the denominator and is the finished answer.
