A radical asks a question: what number, multiplied by itself, gives this? Simplifying one means pulling out every perfect square hiding inside it — a skill Nonlinear Functions and Right Triangles and Trigonometry both assume you already have.
√x means “the number that, multiplied by itself, equals x.” √25 = 5 because 5 × 5 = 25. Most numbers under a radical aren’t perfect squares, though — √50 isn’t a whole number. Simplifying a radical means rewriting it so the only thing left inside the root is a number with no perfect square factors left to pull out.
The perfect squares worth memorizing, since every simplification starts by spotting one of these hiding inside a bigger number:
To simplify a radical like √72, find the largest perfect square that divides evenly into 72, split the radical into two pieces at that factor, then simplify the perfect-square piece. Work through it step by step below.
Any number under a radical can be broken into a perfect-square part and a leftover part — simplifying just means separating the two and pulling the perfect square out from under the root.
First, find the largest perfect square that divides evenly into 72.
Adding and subtracting radicals works exactly like combining like terms — you can only combine two radicals if the number underneath the root (the radicand) is identical. 2√3 + 5√3 = 7√3, the same way 2x + 5x = 7x. Subtraction works the same way: 7√5 − 3√5 = 4√5. But 3√5 + 2√3 can’t be combined at all; √5 and √3 are different, unrelated quantities, just like x and y — a mismatched radicand means the terms simply sit side by side, unsimplified.
Multiplying radicals works the opposite way from adding — you don’t need matching radicands at all. Multiply the coefficients together, multiply the radicands together, then simplify whatever’s left. Skipping steps here is where mistakes creep in, so work it all the way through:
Dividing radicals mirrors multiplication: divide the coefficients separately, divide the radicands separately, then simplify. It’s often faster to simplify first if the radicands share a factor, rather than dividing then simplifying at the end:
One more wrinkle: a radical is only considered fully simplified if there’s no radical left in the denominator. A radical in the numerator is perfectly fine on its own — √8/2 just simplifies normally to 2√2/2 = √2, nothing special required. But 3/√2 needs an extra step, called rationalizing the denominator: multiply the fraction by a form of 1 (the radical over itself) that clears the radical from the bottom without changing the fraction’s value.
A fraction with a radical in the denominator isn't considered simplified. Multiplying by (radical over itself) equals multiplying by 1 — it changes the form without changing the value, and clears the radical from the bottom in the process.
A radical left in the denominator isn't considered simplified. To clear it from 3/√2, what should you multiply by?