Simplifying a square root
A square root is in simplest radical form when no perfect square (4, 9, 16, 25, …) divides the number under the root. To simplify √72:
- Break 72 into prime factors: 72 = 2 × 2 × 2 × 3 × 3.
- Circle the pairs: (2 × 2) and (3 × 3). Each pair comes out as one number: 2 and 3.
- Multiply what came out, and leave the rest inside: √72 = 2 × 3 × √2 = 6√2.
Check: 6√2 ≈ 6 × 1.41421 = 8.48528, and √72 ≈ 8.48528. For cube roots, look for groups of three instead of pairs: ∛54 = ∛(27 × 2) = 3∛2.
Perfect squares to know
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. If a number is between two of them, its square root is between their roots: √50 is between √49 = 7 and √64 = 8, about 7.0711.
Adding square roots
You can only add square roots with the same number inside, like adding like terms: 3√2 + 5√2 = 8√2. Simplify first and you may find matches: √8 + √18 = 2√2 + 3√2 = 5√2.
Square roots of negative numbers
No real number times itself gives a negative, so √−9 isn't a real number. Using the imaginary unit i = √−1, √−9 = 3i. Odd roots of negatives are fine: ∛−8 = −2.