Simplify Radicals Calculator
√72 is not 8.485. That is a rounded decimal, and writing it throws away the exact answer — which is 6√2.
This simplify radicals calculator gives the exact form and shows where it came from: 72 is 36 × 2, the 36 is a perfect square, and its root comes out as 6. The 2 has nothing square about it, so it stays where it is.
How to Simplify a Radical
The whole method is one question asked repeatedly: what is the largest perfect square hiding in there?
Split the number into factors and look for a square. √50 = √(25 × 2) = 5√2. √12 = √(4 × 3) = 2√3. If nothing square divides it, as with √7, then it is already as simple as it gets and there is nothing to do.
Radical expressions with letters work the same way, only easier to see. Under a square root, every pair of a letter comes out as one: √(x⁵) has two pairs and a leftover, so it is x²√x. Under a cube root you look for groups of three instead, and under a fourth root, groups of four.
Both together is just the two ideas at once: √(72x⁵) = 6x²√(2x).
Why √(x²) Is |x| and Not x
This is the part worth slowing down for, because almost nothing else explains it properly.
A square root is never negative. So put x = −3 into √(x²): x² is 9, and √9 is 3 — not −3. Writing √(x²) = x would claim the answer is −3, which is simply wrong. The bars fix it: |−3| = 3.
But the bars are not always needed, and adding them everywhere is its own mistake. Three cases let you drop them:
An odd root. ∛(x³) = x with no bars, because a cube root keeps the sign — ∛(−8) really is −2.
An even power coming out. √(x⁴) = x², no bars, because x² cannot be negative in the first place.
Something stays inside. √(x³) = x√x has no bars either. For that radical to exist at all, x³ must be positive, so x was already positive. The bars would be true but pointless.
The calculator works out which case applies and only adds bars where they carry weight.
How to Use This Simplify Radicals Calculator
Pick the root at the top — square, cube, fourth or fifth — then type what sits underneath and press =. Powers go in with ^, so 72x^5 is what you want for √(72x⁵), and switching the root re-answers straight away.
The answer comes back exact. One tap opens what came out and what stayed, with the perfect power named, and under that the working step by step — which is what simplifying radicals by hand looks like written down. An even root of a negative number is refused rather than guessed at, because there is no real answer to give.
A radical is a fractional power in disguise — √x is x to the power one half — so the exponents calculator handles the same expressions written the other way. It is free to use online, with no account.
Simplifying Radicals FAQ
How do you simplify a radical?
Why is the square root of x squared written as |x|?
When do you not need absolute value bars?
Is the square root of 72 equal to 8.485?
Can you simplify the square root of a sum?
Find the largest perfect power, bring its root out, leave the rest under the sign — and add the bars only when the letter could still be negative.