Radical Operations: Step-by-Step Guide
Radicals (square roots, cube roots, and so on) follow rules a lot like exponents do. You can multiply and divide radicals that have the same root index (like two square roots). But you can only add or subtract radicals that are 'like' radicals, meaning the same root and the same number under the root once they are simplified. It works the same way as 3x + 5x = 8x.
Math Racer skill guide · Algebra 2 · Grade 11
How to solve it, step by step
- Simplify each radical first. Pull out any perfect squares (for square roots) or perfect cubes (for cube roots) from under each root.\[\sqrt{a^{2}b} = a\sqrt{b}\]
- Add or subtract like radicals. Radicals are 'like' if they have the same root and the same number under it. Add or subtract the numbers in front, and keep the radical the same.\[p\sqrt{a} + q\sqrt{a} = (p+q)\sqrt{a}\]
- Multiply radicals. Multiply the numbers in front of the roots together, and multiply the numbers under the roots together. Then simplify.\[\sqrt[n]{a}\,\sqrt[n]{b} = \sqrt[n]{ab}\]
- Divide radicals. Put both numbers under one root as a fraction, simplify, and get rid of any root in the bottom if needed.\[\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\tfrac{a}{b}}\]
Worked example
Simplify: 3 sqrt(8) + sqrt(50) - sqrt(18)
Step 1: Simplify each radical. The square root of 8 = the square root of (4 x 2) = 2 times the square root of 2. The square root of 50 = the square root of (25 x 2) = 5 times the square root of 2. The square root of 18 = the square root of (9 x 2) = 3 times the square root of 2.
Step 2: Put them in: 3(2 root 2) + 5 root 2 - 3 root 2.
Step 3: Multiply the first one: 6 root 2 + 5 root 2 - 3 root 2.
Step 4: Now they are all like radicals (all are root 2). Combine the numbers in front: (6 + 5 - 3) root 2 = 8 root 2.
Step 5: The answer is 8 times the square root of 2.
Check: 3 x root 8 is about 8.485, root 50 is about 7.071, and root 18 is about 4.243. 8.485 + 7.071 - 4.243 = 11.313. And 8 x root 2 is about 11.314. They match.
Tips and common mistakes
- You cannot turn root a + root b into root (a + b). This is a very common mistake.
- Always simplify first. Radicals that look different may turn out to be 'like' radicals.
- Multiplying works as root a x root b = root (ab), as long as a and b are not negative.
- For higher roots, look for perfect cubes (8, 27, 64) or perfect fourth powers (16, 81) under the root.
- Multiply the numbers in front with each other, and the numbers under the roots with each other.
- Do not leave a root in the bottom of a fraction. Rationalize it.
- If the rules feel awkward, write the root as a power: the nth root of a is a^(1/n).