Nth Root Calculator

What is an nth root?

The nth root of x is the number r with rⁿ = x. Taking a root undoes raising to a power: 2⁵ = 32, so the 5th root of 32 is 2. It is written with a radical sign, ⁿ√x, where n is the index and x is the radicand (the number under the sign). An index of 2 is usually left out, so √x means the square root. The table names the common roots.

Roots as powers

A root is the same as a fractional power: ⁿ√x = x^(1/n). So ∛8 = 8^(1/3) = 2, and √9 = 9^(1/2) = 3. A root of a power combines the two: ⁿ√(x^m) = x^(m/n), so 8^(2/3) = (∛8)² = 4. This is also how to find a root on most calculators and spreadsheets, as shown further down.

Perfect powers

A number is a perfect nth power when its nth root is a whole number. These are worth knowing by sight, because they tell you at once which whole numbers a root lies between.

Swipe the table sideways to see 4th and 5th powers.

How to find the nth root with prime factors

Prime factors give the exact answer and show whether it is a whole number. Break the number into primes, then make groups of n equal factors. Each full group comes out of the root as one factor; anything left over stays inside. For the cube root of 54:

1. Factor 54: 54 = 2 · 3 · 3 · 3 = 2 · 3³.
2. The three 3s make one group of three, so a 3 comes out. The 2 has no partners, so it stays inside.
3. Write the result: ∛54 = 3∛2 ≈ 3.7798.
4. Check by cubing it: (3∛2)³ = 27 · 2 = 54.

Show ∛54 step by step

Square roots work the same way with pairs: 50 = 2 · 5², so √50 = 5√2 ≈ 7.0711. Sometimes the index can shrink as well. 64 = 2⁶, so ∜64 = 2∜(2²), and the 4th root of 2² is just √2. That gives ∜64 = 2√2 ≈ 2.8284, which is the simplest radical form.

Show ∜64 in simplest radical form

Estimating a root and finding it by hand

Start with the perfect powers on either side. 54 lies between 3³ = 27 and 4³ = 64, so ∛54 is between 3 and 4. 54 is much closer to 64 than to 27, so the root is closer to 4.

To get more digits, try one decimal place at a time and see whether you went over: 3.7³ = 50.653 is too small and 3.8³ = 54.872 is too big, so the root starts 3.7. Next, 3.77³ ≈ 53.583 and 3.78³ ≈ 54.01, so it starts 3.77. Each round adds one correct digit. The curve in the calculator shows the same idea: the point where the line from your number meets y = xⁿ.

Roots of fractions and decimals

The root of a fraction is the root of the top over the root of the bottom. When both are perfect powers the answer is a fraction: ∛(8/27) = ∛8 / ∛27 = 2/3.

Show the cube root of 8/27

When the bottom is not a perfect power, multiply top and bottom so that it becomes one. For √(1/2), multiply by 2/2: √(2/4) = √2 / 2 ≈ 0.7071. Decimals become fractions first: 0.125 = 1/8, so ∛0.125 = 1/2 = 0.5.

Roots of negative numbers

An odd root of a negative number is negative, because a negative number multiplied by itself an odd number of times stays negative. (−5)³ = −125, so ∛−125 = −5.

Show the cube root of −125

An even root of a negative number has no real value. Any real number to an even power is 0 or positive, so nothing real gives −81 when raised to the 4th power. The answer exists only as complex numbers: the four 4th roots of −81 are 3 in size and point at 45°, 135°, 225° and 315°, which is about ±2.1213 ± 2.1213i.

Show the 4th roots of −81

How many nth roots does a number have?

Counting complex numbers, every number other than 0 has exactly n nth roots. They all have the same size and sit evenly around a circle, 360°/n apart. How many of them are real depends on n and the sign:

• A positive number has two real roots for an even n (16 has 4th roots 2 and −2) and one for an odd n.
• A negative number has one real root for an odd n (−8 has cube root −2) and none for an even n.
• 0 has just one root, 0.

The symbol ⁿ√x means the principal root: the positive one for a positive x, and the real one for an odd root of a negative x. So √16 = 4, even though −4 also squares to 16. The 4th roots of 16 are 2, 2i, −2 and −2i, and the calculator lists all of them.

Show all four 4th roots of 16

Rules for working with roots

• The root of a product is the product of the roots: ⁿ√(ab) = ⁿ√a · ⁿ√b. That is the rule behind simplifying.
• The root of a quotient is the quotient of the roots: ⁿ√(a/b) = ⁿ√a / ⁿ√b.
• A root and a power cancel: (ⁿ√x)ⁿ = x. For even n, ⁿ√(xⁿ) = |x|, so √((−3)²) = 3.
• A root of a root multiplies the indexes: √(∛x) = ⁶√x.
• There is no rule for sums: √(9 + 16) = 5, but √9 + √16 = 7.

Worked examples

Homework often says "find the indicated root": the index tells you which root to take. Check each answer by raising it to the index; you should get the number under the root back. Each row opens in the calculator with the steps.

The last two rows are not perfect powers, so part of the root stays inside. 16 is 2³ times 2, so one 2 comes out of the cube root. 72 is 6² times 2, so a 6 comes out of the square root. In decimals, ∛16 ≈ 2.5198 and √72 ≈ 8.4853.

Common irrational roots

Most roots are irrational: their decimals never end and never repeat, so any decimal you write is rounded. These come up often enough to be worth recognizing. Each value is rounded to 6 decimal places.

Where nth roots are used

• From a volume to a side. A cube that holds 1,000 cm³ has sides of ∛1000 = 10 cm. A square room of 20 m² has walls √20 ≈ 4.47 m long.
• Average growth rates. If an investment grows from 100 to 150 in 5 years, the yearly growth factor is ⁵√(150/100) = ⁵√1.5 ≈ 1.0845, about 8.45% a year. The Geometric Mean calculator works this out as a CAGR.
• The geometric mean. The geometric mean of n numbers is the nth root of their product: for 2, 8 and 4 it is ∛64 = 4.
• Music. In equal temperament the octave (a frequency ratio of 2) is split into 12 equal steps, so each semitone multiplies the frequency by ¹²√2 ≈ 1.0595.

Finding an nth root on a calculator or in a spreadsheet

Any calculator with a power key can find an nth root by raising to the power 1/n. Keep the brackets around 1/n:

Without the brackets, 54^1/3 means 54¹ ÷ 3 = 18. Many scientific calculators also have an x-th root key, often the second function of the power key. On Casio fx-991 models, press SHIFT and then the power key (x^■) to get the root template: type the index in the small box, press the right arrow, type the number and press =. In Excel and Google Sheets, type =POWER(54, 1/3) or =54^(1/3). Spreadsheets return an error for a fractional power of a negative number, so for an odd root of a negative write the minus outside: =-POWER(125, 1/3) gives −5.

Common mistakes

• Dividing by n instead of taking the root. The cube root of 27 is 3, not 9.
• Typing 27^1/3 without brackets, which gives 9.
• Saying an even root of a negative number is negative. ∜−16 is not −2, because (−2)⁴ = 16.
• Leaving a perfect power inside the root: √50 should be 5√2.
• Splitting a root over a sum. √(a + b) is not √a + √b.