Undoing a power
A square root asks which number, multiplied by itself, gives this one: √49 = 7 because 7 × 7 = 49. A cube root asks the same with three copies: ∛27 = 3 because 3 × 3 × 3 = 27. Roots turn up in area and volume problems, in distances, in statistics and throughout algebra.
This square root and cube root calculator gives the root to eight decimal places, tells you whether the number is a perfect square or cube, and for whole numbers writes the simplified radical, the form maths teachers ask for, such as √72 = 6√2. The third mode finds any root, from fourth roots upward.
How to use it
- Choose Square root, Cube root or Any root.
- Type the number. For Any root, also say which root: 4 for a fourth root, 5 for a fifth.
- Read the root, the check, and the simplified radical if the number is whole.
- Negative numbers work with odd roots, such as cube roots; even roots of negatives are not real numbers.
The root formulas
√x = x^(1/2); ∛x = x^(1/3); nth root of x = x^(1/n)
simplify: √(a² × b) = a√b, for example √72 = √(36 × 2) = 6√2A root is a fractional power, which is why x^(1/2) and √x are the same thing. To simplify a radical, find the largest perfect square (or cube) that divides the number, take its root outside, and leave the rest under the sign.
Worked example: √72
- √72 = 72^(1/2) = 8.48528137
- 72 is not a perfect square: 8² is 64 and 9² is 81
- Largest square factor: 36, and 72 = 36 × 2
- So √72 = √36 × √2 = 6√2
The square root of 72 is 8.48528137, which simplifies to 6√2. A cube root works the same way with cubes: 54 = 27 × 2, so ∛54 = 3∛2.
Roots to know
| Number | Square root | Cube root |
|---|---|---|
| 2 | 1.4142 | 1.2599 |
| 3 | 1.7321 | 1.4422 |
| 5 | 2.2361 | 1.71 |
| 10 | 3.1623 | 2.1544 |
| 50 | 7.0711 | 3.684 |
| 100 | 10 | 4.6416 |
| 1000 | 31.6228 | 10 |
Perfect squares up to 144 are worth memorising: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144. They make estimating easy: √50 sits just above 7, because 49 is 7².
Common slips
- Forgetting that every positive number has two square roots, +7 and −7 for 49. The √ sign means the positive one.
- Splitting a sum: √(9 + 16) is 5, not √9 + √16 = 7.
- Taking an even root of a negative number. It has no real answer; it needs complex numbers.
- Rounding too early. Keep full precision until the last step, then round.
- For powers and roots together, the scientific calculator handles full expressions.
Frequently asked questions
How do I find a square root without a calculator?
Find the perfect squares on either side. For √50, 49 is 7² and 64 is 8², so the answer is just over 7. Refine by trying 7.1: 7.1² is 50.41, a little high, so √50 is about 7.07. Enter the number above for the exact value to eight decimal places.
How do I simplify a square root?
Find the largest perfect square that divides the number, take its root outside the sign, and leave the rest inside. For √72, the largest square factor is 36, so √72 = √36 × √2 = 6√2. The calculator does this for any whole number up to a trillion.
What is the cube root of a negative number?
It is negative. Because (−3) × (−3) × (−3) = −27, the cube root of −27 is −3. This works for any odd root. Even roots of negative numbers, such as √−4, have no real answer, and the calculator tells you so instead of giving a wrong number.
What is a perfect square?
A perfect square is a whole number that is another whole number times itself, such as 16 = 4 × 4 or 144 = 12 × 12. Its square root is a whole number. The calculator checks this for you, and does the same for perfect cubes and higher powers.
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