Simplifying Radicals: Step-by-Step Guide
A radical (like a square root) is in simplest form when nothing under the root sign can be pulled out, there are no fractions under the root, and there are no roots in a denominator. To simplify, look for perfect squares (or perfect cubes, and so on) hiding inside the root and pull them out.
Math Racer skill guide · Algebra 2 · Grade 11
How to solve it, step by step
- Factor the number under the root. Break the number into factors. Look for the biggest perfect square (like 4, 9, 16, 25, 36) that divides it.\[\sqrt{n} = \sqrt{p\cdot q}\;\text{where }p\text{ is a perfect square}\]
- Split the root. Use the rule: the square root of (a x b) = the square root of a times the square root of b. This splits the perfect square from the rest.\[\sqrt{p\cdot q} = \sqrt{p}\,\sqrt{q}\]
- Take the root of the perfect square. The root of the perfect square comes out front as a whole number. The rest stays under the root sign.\[\sqrt{a^{2}} = a\;(a \geq 0)\]
- Check that you are finished. Make sure no perfect square is left under the root, there is no fraction under it, and there is no root in a denominator.\[\sqrt{n} = a\sqrt{b}\;\text{with }b\text{ square-free}\]
Worked example
Simplify: sqrt(72)
Step 1: Look for the biggest perfect square that divides 72. It is 36, because 72 = 36 x 2.
Step 2: Split the root: the square root of 72 = the square root of (36 x 2) = the square root of 36 x the square root of 2.
Step 3: The square root of 36 is 6. So the square root of 72 = 6 times the square root of 2.
Step 4: Check: 2 is prime, so no more perfect squares can come out.
Step 5: The answer is 6 times the square root of 2.
Check: (6 x the square root of 2)^2 = 36 x 2 = 72. It matches.
Tips and common mistakes
- Memorize the perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
- Pull out the LARGEST perfect square. If you use 4 when 36 is available, you will have more work to do.
- If you cannot spot the biggest factor, break the number into primes. Each pair of the same prime comes out as one.
- For letters: the square root of x^n is x^(n/2) when n is even. When n is odd, pull out x^((n-1)/2) and leave one x inside.
- For cube roots, look for groups of three of the same prime. For fourth roots, groups of four.
- Unless the problem says otherwise, assume the letters stand for positive numbers. Otherwise, the square root of x^2 is |x|.
- If there is a fraction under the root, split it: the square root of (a/b) = the square root of a over the square root of b.
- Check by squaring your answer. You should get back the number you started with.