How to Calculate a Square Root
The square root of a number x is the value that, multiplied by itself, gives x. Every positive number has two square roots, one positive and one negative; the symbol √x means the positive (principal) root. For example, √49 = 7 because 7 × 7 = 49. More generally, the nth root of x is the number that gives x when raised to the power n.
√x = x1/2 ∛x = x1/3 ⁿ√x = x1/n
How to Simplify a Square Root
To write a radical in simplest form, factor the number and pull out every factor that appears in a complete group of n (pairs for square roots, triples for cube roots).
Worked Example: Simplify √72
- Prime factorization: 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²
- Pairs: one pair of 2s and one pair of 3s come out as 2 × 3 = 6
- Left inside: a single 2
- Result: √72 = 6√2 ≈ 8.485281374
Cube root example: 54 = 3³ × 2, so ∛54 = 3∛2 ≈ 3.77976315.
Perfect Squares from 1 to 225
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. Their square roots are the whole numbers 1 through 15.
Roots of Negative Numbers
Odd roots of negative numbers are real: ∛−27 = −3. Even roots of negative numbers are not real numbers. For square roots, the calculator shows the imaginary result using i = √−1, so √−16 = 4i.
Frequently Asked Questions
How do you simplify a square root?
Factor the number, take out each pair of equal factors as a single factor outside the radical, and leave the rest inside. For example, √72 = √(36 × 2) = 6√2.
What is the square root of 2?
The square root of 2 is an irrational number, approximately 1.414213562. It cannot be written exactly as a fraction.
Can you take the square root of a negative number?
Not within the real numbers. In complex numbers, √−x = i√x, where i is the imaginary unit. For example, √−9 = 3i.
What is the difference between a square root and a cube root?
A square root multiplied by itself gives the original number (√25 = 5 because 5² = 25). A cube root multiplied by itself three times gives the original number (∛27 = 3 because 3³ = 27).
How do I calculate an nth root on a calculator?
Raise the number to the power 1/n. For example, the 4th root of 81 is 81^(1/4) = 3.