Square roots and cube roots, and how to find them
What square, cube and nth roots mean, how to simplify a surd such as √72, how to find a root by hand with prime factors or by estimation, and how roots appear in real problems.
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In short
- The square root of a number is the value that, multiplied by itself, gives that number. √49 = 7 because 7 × 7 = 49.
- A cube root is multiplied by itself three times in all. ∛27 = 3 because 3 × 3 × 3 = 27.
- To simplify a surd, pull out the largest perfect square. √72 = √36 × √2 = 6√2.
- Square roots of negative numbers are not real numbers, but cube roots of negatives are. ∛−8 = −2.
- Most square roots go on forever without repeating, so any decimal is an approximation.
A root undoes a power. If squaring takes 7 to 49, the square root takes 49 back to 7. Roots appear in school algebra, in the Pythagorean theorem, in standard deviation, in compound interest and whenever an area is known but a side is wanted. This page explains square, cube and higher roots, shows how to simplify a surd the way exams expect, and sets out the methods for finding a root without a calculator.
What a root means
√x = y means y × y = x
∛x = y means y × y × y = x
More generally, the nth root of x is the number that gives x when used n times in a multiplication. The fifth root of 32 is 2, because 2⁵ = 32. In exponent form, the nth root of x is x raised to the power 1/n, so √x = x^(1/2).
A perfect square is a whole number whose square root is also whole, such as 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. A perfect cube is the same idea for cube roots, such as 1, 8, 27, 64, 125 and 216.
Strictly, 49 has two square roots, 7 and −7, as (−7) × (−7) is also 49. The symbol √ means the positive one, called the principal square root.
Simplifying a surd
A surd is a root left in root form because its decimal never ends. Exams usually want the simplest form rather than a decimal.
To simplify, find the largest perfect square that divides the number and pull its root outside:
√(a² × b) = a√b
Take √72. The largest perfect square dividing 72 is 36, so √72 = √36 × √2 = 6√2.
Using prime factors is a reliable way to do this. Write 72 = 2 × 2 × 2 × 3 × 3. Each pair of the same prime moves outside as one copy: one pair of 2s and one pair of 3s give 2 × 3 = 6. A single 2 stays inside, which gives 6√2.
| Number | Simplified | Decimal |
|---|---|---|
| √8 | 2√2 | 2.8284 |
| √12 | 2√3 | 3.4641 |
| √18 | 3√2 | 4.2426 |
| √50 | 5√2 | 7.0711 |
| √72 | 6√2 | 8.4853 |
| √98 | 7√2 | 9.8995 |
The calculator shows this simplified form for any whole number.
Finding a root by hand
Prime factors, for perfect squares and cubes. To find √1764, factorise it: 1764 = 2² × 3² × 7². Halve each power: 2 × 3 × 7 = 42. For a cube root, divide each power by 3 instead. For 1,728 = 2⁶ × 3³, the cube root is 2² × 3 = 12.
Estimation. To estimate √50, find the perfect squares around it. It lies between √49 = 7 and √64 = 8, and very close to 7. Try 7.1: 7.1² = 50.41, which is a little over. So √50 is just under 7.1, and in fact about 7.07.
Averaging. A quick way to improve an estimate is to divide the number by your guess, then average the guess and the result. For √50, guess 7: 50 ÷ 7 = 7.14, and the average of 7 and 7.14 is 7.07. One round is often enough, and each further round roughly doubles the correct digits. This is an old method known as the Babylonian or Heron's method.
Long division. Indian school books teach a long-division method for square roots that gives one digit at a time. It is slow but reliable, and useful when the question asks for it.
Rules worth remembering
√(a × b) = √a × √b
√(a ÷ b) = √a ÷ √b
But roots do not split over addition. √(a + b) is not √a + √b. For example, √(9 + 16) = √25 = 5, while √9 + √16 = 3 + 4 = 7. This is one of the most common algebra mistakes.
Roots of decimals. Move the decimal point two places at a time for square roots. √0.25 = 0.5, as 0.5² = 0.25. Notice that the root of a number between 0 and 1 is larger than the number itself.
Negative numbers. Any real number squared is positive or zero, so a negative number has no real square root. The calculator says so. Odd roots are fine, as −2 × −2 × −2 = −8, so ∛−8 = −2.
Irrational roots
Roots of non-perfect squares are irrational: they cannot be written as a fraction, and their decimals never end or repeat. √2 starts 1.41421356 and goes on forever. The calculator rounds to eight decimal places, which is far more than any practical use requires.
The fact that √2 is irrational was one of the first surprising discoveries in Greek mathematics. It is also the length of the diagonal of a square with sides of 1.
Where roots are used
Geometry. The side of a square of area 144 is √144 = 12. A plot of 2,500 sq ft, if it is square, has sides of 50 ft. The area calculator goes the other way. The hypotenuse of a right triangle comes from a square root too, as the Pythagorean theorem calculator shows.
Volume. The side of a cube that holds 1 m³ is ∛1 = 1 m, and a cubic tank of 8 m³ has sides of 2 m.
Statistics. Standard deviation is the square root of the variance.
Finance. Average growth over n years uses the nth root. If money doubles in 10 years, the yearly growth is the tenth root of 2 minus 1, about 7.18 percent, which is what the CAGR calculator works out.
Physics. Fall time, pendulum periods and many other formulas involve a square root.
Roots worth knowing
A few values come up so often that they are worth memorising, because many others follow from them.
| Root | Value |
|---|---|
| √2 | 1.4142 |
| √3 | 1.7321 |
| √5 | 2.2361 |
| √6 | 2.4495 |
| √7 | 2.6458 |
| √10 | 3.1623 |
With these, you can reach other values quickly. Since √200 = √100 × √2, it is 14.142. Likewise, √0.02 = √2 ÷ 10 = 0.14142, and √75 = 5√3 = 8.660.
Shortcuts for perfect squares and cubes
The last digit of a square. A perfect square never ends in 2, 3, 7 or 8. So 4,332 cannot be a perfect square, and you can tell at a glance.
Finding a square root from the ends. Take 1,764. It ends in 4, so its root ends in 2 or 8. Cover the last two digits to leave 17, which lies between 4² = 16 and 5² = 25, so the root is in the forties. That leaves 42 or 48. Since 45² = 2,025 is above 1,764, the root is below 45, and the answer is 42.
Finding a cube root from the ends. The last digit of a perfect cube fixes the last digit of its root: 1 gives 1, 8 gives 2, 7 gives 3, 4 gives 4, 5 gives 5, 6 gives 6, 3 gives 7, 2 gives 8, and 9 gives 9. For 74,088, the root ends in 2. Cover the last three digits to leave 74, which lies between 4³ = 64 and 5³ = 125. So the root is 42.
These tricks work only when the number is a perfect square or cube. The calculator tells you whether it is.
What this calculator assumes
- The root must be a whole number of 2 or more.
- An even root of a negative number is reported as not real, rather than as a complex number.
- Results are rounded to eight decimal places, using ordinary computer arithmetic, which is accurate for normal numbers.
- The simplified surd and the perfect-square check are shown only for positive whole numbers.
- The nearest perfect squares are listed to help with estimating by hand.