Root Calculator

1. Enter the radicand a. This is the number inside the radical. It can be negative, a decimal or in e notation.
2. Enter the index n. 2 is a square root, 3 a cube root, and any larger whole number works up to 1000.
3. Read the root and the checks. The tool tests for a perfect power, simplifies the radical for whole numbers, shows Newton steps and raises the answer back to the nth power.

An nth root undoes raising to the nth power. OpenStax Intermediate Algebra 2e states it directly: if b^n = a, then b is an nth root of a, and n is called the index of the radical.

The index decides how negatives behave. The same OpenStax section lists the properties: with an even index, a root of a negative number is not a real number; with an odd index, the root is real for every value of a. So the 5th root of -32 is -2, but the 4th root of -16 has no real value.

An even index also brings a sign question. Both 3 and -3 give 81 when raised to the 4th power, so the equation y^4 = 81 has two real solutions. The radical symbol names only one of them, the principal root, which OpenStax College Algebra 2e defines as the nonnegative one. The 4th root of 81 is 3, not plus or minus 3.

Square roots and cube roots are the n = 2 and n = 3 cases. For those alone, the cube root calculator and the square root tool go deeper; this page handles every index.

The tool computes the root as a power with exponent 1/n, snaps it to an exact value when the radicand is a perfect power, and simplifies the radical by prime factors.

1. Check the sign: an even index with a negative radicand has no real root.
2. Take |a|^(1/n) and give it the sign of a when n is odd.
3. Test nearby whole numbers: if one raised to n equals a exactly, the root is exact.
4. For whole numbers, factor a into primes and move every group of n equal factors outside the radical.
5. Raise the answer back to the nth power as a check.

OpenStax Intermediate Algebra 2e calls a radical simplified when the radicand has no factor that is an nth power. The Simplified radical row applies exactly that test.

A Perfect Power: the 5th Root of 243

3 x 3 x 3 x 3 x 3 = 243, so the 5th root of 243 is exactly 3.

A Negative Radicand: the 5th Root of -96

1. The index 5 is odd, so a real root exists and it is negative.
2. Factor: 96 = 2^5 x 3. The group 2^5 comes out as 2.
3. Simplified: -2 root5(3).
4. Decimal: -2 x 1.24573094 = -2.49146188.

A Root That Never Ends: the 6th Root of 10

1^6 = 1 and 2^6 = 64, so the root lies between 1 and 2. Newton steps from 1.5 give 1.469479, 1.467804 and 1.467799. The value is 1.46779927 to 8 places, and 1.46779927^6 returns 10 to that precision.

The index, the sign of the radicand and its size control what kind of answer you get.

Even vs Odd Index

An even index needs a radicand of zero or more for a real answer. An odd index accepts any real number and keeps its sign.

Radicand Between 0 and 1

Roots of numbers between 0 and 1 are larger than the number itself. The 4th root of 0.0081 is 0.3, because 0.3^4 = 0.0081.

Larger Index

As n grows, the root of any number above 1 shrinks toward 1. The 10th root of 1024 is 2, and the 12th root of 2 is about 1.05946309.

Perfect Powers

Only perfect nth powers have whole-number or terminating roots. Every other positive whole number has an irrational root with a decimal that never repeats.

The pattern follows the index, not the size of the number. To go the other way and raise a number to a power, use the exponent calculator.

Solving Power Equations

An equation such as y^5 = 243 is solved by taking the 5th root: y = 3. For an even power, remember both signs: y^4 = 81 gives y = 3 or y = -3.

Average Growth Rates

The compound growth rate over n periods uses an nth root: (end / start)^(1/n) - 1. Growth from 100 to 200 over 6 years is 2^(1/6) - 1 = 12.25% a year.

Music Tuning

In twelve-tone equal temperament each semitone multiplies the frequency by the 12th root of 2, about 1.05946309. The note frequency calculator uses this ratio.

Geometry and Scaling

The side of a square comes from a square root of its area, and the side of a cube from a cube root of its volume. Higher roots appear when a quantity scales with a higher power.

1. Calling the Principal Root Plus or Minus

The 4th root of 81 is 3. The plus-or-minus answer belongs to the equation y^4 = 81, not to the radical itself.

2. Rejecting Every Negative Radicand

Only even roots of negatives fail. The cube root of -8 is -2 and the 7th root of -1 is -1.

3. Dividing by N Instead of Taking a Root

The 4th root of 16 is 2, not 16 / 4 = 4. A root is an exponent of 1/n, not a division by n.

4. Pulling Out the Wrong Factor

Only factors that are perfect nth powers come out. For the 4th root of 48 the factor is 16 = 2^4, giving 2 root4(3), not 4 root4(3).

5. Typing a Negative in Front of a Calculator Power

Some calculators read -32^(1/5) as -(32^(1/5)) and others reject (-32)^(1/5). This tool uses the real-root rule for odd indexes.

6. Rounding Too Early

Rounding a root before using it in a longer calculation adds error. Keep more places, then round the final answer. For very large or small values, the scientific notation calculator helps keep track of magnitude.

Perfect powers return exact roots; every other root is rounded to the decimal places you choose, with about 15 significant digits available.

It is 2, because 2 x 2 x 2 x 2 = 16. The equation y^4 = 16 also has the solution -2.

Yes. Any odd root of a negative number is real and negative. The 3rd root of -125 is -5.

No real number squared gives a negative result. The answer exists only among complex numbers: sqrt(-16) = 4i.

Use the power 1/n. The 5th root of 32 is 32^(1/5) = 2.

It is 1 for every index. For an even index, -1 also solves y^n = 1.

It is 0 for every index, because 0 raised to any positive power is 0.

Write 54 = 27 x 2 = 3^3 x 2, then pull out the 3: the result is 3 cbrt(2).

It means the cube root of 8, squared: 2^2 = 4. The denominator is the root and the numerator is the power.

It is correct to about 15 significant digits and shown to the places you choose. The check row confirms it.

No. The math runs in your browser. Save keeps a copy only in your own browser storage.

• Nth roots, the index and the even and odd rules (OpenStax Intermediate Algebra 2e, section 8.1).
• When a radical expression is simplified (OpenStax Intermediate Algebra 2e, section 8.2).
• The principal root and rational exponents (OpenStax College Algebra 2e, section 1.3).