Enter which root you want (the index n) and the number under the root (a). The math expression below is linked to the input fields, so you can also edit the small index or the number under the root directly. An index of 2 gives the square root (√a), and 3 gives the cube root (∛a).
The index n can be any number except 0 (decimals such as 2.5 and negative numbers work too). The number under the root a can be negative only when the index n is a positive odd whole number (1, 3, 5, ...).
Result
Enter which root you want (the index) and the number in the fields on the left and press "Calculate". The result will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formula
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter which root you want (the index \(n\)) and the number under the root (the radicand \(a\)). Square roots such as \(\sqrt{27}\), cube roots and any \(n\)th root are all found on this one page
- An index of 2 gives the square root (\(\sqrt{a}\)), and 3 gives the cube root (\(\sqrt[3]{a}\)). Decimal indexes such as 2.5 and negative indexes also work
- Odd roots of negative numbers, such as \(\sqrt[3]{-8} = -2\), are calculated as they are
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
A positive number has two square roots, \(+\sqrt{a}\) and \(-\sqrt{a}\). This calculator shows the value of the radical, which is the positive one (the principal root). Even roots of negative numbers, such as the square root of a negative number, do not exist among the real numbers, so they cannot be calculated.
What is this calculation used for?
Finding a side length from an area or a volume (home and DIY)
For a square garden bed with an area of 27 square feet, each side is \(\sqrt{27} \approx 5.2\) feet. For a cube-shaped box that holds 1 cubic foot (\(1{,}728\) cubic inches), each edge is \(\sqrt[3]{1728} = 12\) inches.
Going back from "the result of multiplying" to "the original length" is always a root. It comes up in gardening, storage and choosing a tank, whenever the area or the volume is decided first.
Finding an average annual growth rate (money)
If an investment doubles in 10 years, by what percent did it grow each year? The answer is a 10th root: \(\sqrt[10]{2} \approx 1.0718\), so about 7.2% per year on average.
Fund reports and company revenue growth (CAGR, the compound annual growth rate) are calculated exactly this way, as "the \(n\)th root of the growth over \(n\) years". It lets you check compound growth for yourself and ties directly to money decisions.
Why A4 paper keeps its shape when folded, thanks to \(\sqrt{2}\) (design)
Most of the world uses A-series paper (A3, A4, A5, ...), where the long side is \(\sqrt{2}\) (about 1.414) times the short side. With this ratio, folding a sheet in half gives exactly the same shape, so enlarging or reducing a copy fits perfectly.
US Letter paper (8.5 × 11 in) does not work this way. Its ratio is about 1.29, but half a sheet (5.5 × 8.5 in) has a ratio of about 1.55. Only the ratio \(\sqrt{2}\), "the number whose square is 2", keeps the shape the same.
The piano scale is built on "the 12th root of 2" (music)
A note one octave higher has exactly twice the frequency. Modern instruments split that octave evenly into 12 half steps, so each half step up multiplies the frequency by \(\sqrt[12]{2} \approx 1.0595\) (twelve-tone equal temperament).
Piano tuning and synthesizer sound design both use this "12th root of 2". One step of the key-change button on a karaoke machine is this same ratio in frequency.
The last step of the standard deviation (statistics and quality control)
The standard deviation, used to analyze test scores and in factory quality control, is calculated as "the square root of the variance (the average of the squared deviations)". If the variance is 9 points², the standard deviation is \(\sqrt{9} = 3\) points.
A square root is needed to bring a squared quantity back to the original unit. This calculation runs wherever statistics are used, from the margin of error in polls to the tolerance of a product.
Formula
Definition of the \(n\)th root (the number that gives \(a\) when raised to the power \(n\))
Standard notation
(the usual math form)
\(b\)
\(n\)
\(=\)
\(a\)
In words
(symbols replaced with words)
①
\(b\): the \(n\)th root of \(a\)
②
\(n\): index
\(=\)
③
\(a\): number under the root
The formula in words
①
Multiply
\(b\): the \(n\)th root of \(a\)
by itself
②
\(n\): index
times,
③
and you get back exactly
\(a\): number under the root
Quick example
The cube root of 27 is 3. Checking that "3 multiplied by itself 3 times gives back 27"
cube root of 27 (3)
index (power 3)
\(=\)
number under the root (27)
\(3 \times 3 \times 3 = 27\)
Key idea
"The \(n\)th root of \(a\)" is the number that gives \(a\) when raised to the power \(n\). It is written \(\sqrt[n]{a}\). It is exactly the reverse of raising to a power (an exponent), so think of it as "undoing a power".
The small number written at the upper left of the radical sign is the index \(n\). Only when \(n = 2\) (the square root) is it left out by convention, and we write \(\sqrt{a}\). The root with \(n = 3\), \(\sqrt[3]{a}\), is called the cube root.
Writing a root as an exponent (\(\sqrt[n]{a} = a^{1/n}\))
Standard notation
(the usual math form)
\(\sqrt[n]{a}\)
\(=\)
\(a\)
\(1/n\)
In words
(symbols replaced with words)
③
\(\sqrt[n]{a}\): the \(n\)th root of \(a\)
\(=\)
①
\(a\): number under the root
②
\(\frac{1}{n}\): exponent
The formula in words
①
Raise
\(a\): number under the root
②
to the power of
\(\frac{1}{n}\): exponent (1 over \(n\))
③
and you get
\(\sqrt[n]{a}\): the \(n\)th root of \(a\)
Quick example
Writing \(\sqrt{27}\) (the square root of 27) as an exponent and calculating it
square root of 27, \(\sqrt{27}\)
\(=\)
number under the root (27)
exponent (1/2)
\(27^{1/2} = \sqrt{27} \approx 5.196\)
Key idea
Calculators, Excel and programming languages often have no way to type a radical sign, so roots are calculated in this "power of \(\frac{1}{n}\)" form (for example, \(\sqrt[3]{10} = 10^{1/3}\)). This calculator also uses this formula inside.
We may use a fraction such as \(\frac{1}{n}\) as an exponent because the law of exponents gives \((a^{1/n})^n = a^{n \times (1/n)} = a^1 = a\). That is exactly the definition of the \(n\)th root: raise it to the power \(n\) and you get back \(a\). The index \(n\) does not have to be 2 or 3. Decimals such as 2.5 and negative numbers also make sense as "the power \(\frac{1}{n}\)" (in that case, only for \(a > 0\)).
Roots of negative numbers (only when the index is odd)
Standard notation
(the usual math form)
\(\sqrt[n]{-a}\)
\(=\)
\(-\sqrt[n]{a}\)
In words
(symbols replaced with words)
②
\(\sqrt[n]{-a}\): the \(n\)th root of the negative number \(-a\)
\(=\)
①
\(-\sqrt[n]{a}\): the root with a minus sign
The formula in words
①
When the index \(n\) is odd,
\(-\sqrt[n]{a}\): the \(n\)th root of the positive number \(a\) with a minus sign
is
②
\(\sqrt[n]{-a}\): the \(n\)th root of the negative number \(-a\)
Quick example
For the cube root of \(-8\), first find the cube root of \(8\), \(\sqrt[3]{8} = 2\), then add a minus sign
cube root of −8, \(\sqrt[3]{-8}\)
\(=\)
cube root of 8 (2) with a minus sign
\(\sqrt[3]{-8} = -\sqrt[3]{8} = -2\)
\((-2) \times (-2) \times (-2) = -8\)
Key idea
A negative number multiplied by itself an odd number of times stays negative (for example, \((-2)^3 = -8\)), so negative numbers do have odd roots.
On the other hand, a negative number multiplied by itself an even number of times is always positive (for example, \((-2)^2 = +4\)). So there is no real number whose square is \(-4\), and even roots of negative numbers (such as the square root of a negative number) cannot be calculated among the real numbers. That is why this calculator accepts a negative number under the root only when the index is odd.
Finding a square root by hand (averaging again and again)
Standard notation
(the usual math form)
\(b'\)
\(=\)
\((\)
\(b\)
\(+\)
\(a \div b\)
\()\)
\(\div\)
\(2\)
In words
(symbols replaced with words)
④
\(b'\): new estimate
\(=\)
\((\)
①
\(b\): current estimate
\(+\)
②
\(a \div b\): number under the root ÷ current estimate
\()\)
\(\div\)
③
2 (take the average of the two)
The formula in words
①
Add
\(b\): current estimate
②
and
\(a \div b\): number under the root ÷ current estimate
③
divide by
2 (take the average of the two)
④
and you get
\(b'\): new estimate
(the more you repeat this, the closer you get to the true square root)
Quick example
Improving an estimate of \(\sqrt{27}\) (the answer is about 5.196) once, starting from \(b = 5\)
new estimate
\(=\)
\((\)
current estimate (5)
\(+\)
27 ÷ 5 (= 5.4)
\()\)
\(\div\)
2
\((5 + 27 \div 5) \div 2 = (5 + 5.4) \div 2 = 5.2\)
\(5.2 \times 5.2 = 27.04 \approx 27\)
Key idea
If the estimate \(b\) is smaller than the true square root, then \(a \div b\) is larger than it, and the other way around. The true square root always lies between these two numbers, so taking their average jumps much closer to it. That is the trick behind this method. Repeat the example above once more and you get \((5.2 + 27 \div 5.2) \div 2 = 5.19615\ldots\), which is correct to 5 decimal places.
The way computers and calculators find square roots (Newton's method) is built on the same idea. For an \(n\)th root, a different average, \(b' = \{(n-1) \times b + a \div b^{n-1}\} \div n\), plays the same role.
An nth root finds "the number that gives a when raised to the power n". It is the reverse of raising to a power. To calculate it, the basic move is to rewrite it as "a to the power 1/n". Roots of negative numbers exist among the real numbers only when the index is odd.
Symbols and terms
Symbols
| \(\sqrt{a}\) | square root of a | The square root of \(a\) (the number, 0 or greater, whose square is \(a\)). By convention the small 2 at the upper left of the radical sign is left out. (Example - \(\sqrt{27} \approx 5.196\)) |
| \(\sqrt[3]{a}\) | cube root of a | The number that gives \(a\) when cubed. (Example - \(\sqrt[3]{27} = 3\), \(\sqrt[3]{-8} = -2\)) |
| \(\sqrt[n]{a}\) | nth root of a | The number that gives \(a\) when raised to the power \(n\). The small \(n\) at the upper left of the radical sign is the index (which root it is). |
| \(a^{1/n}\) | a to the power 1 over n | The same value as \(\sqrt[n]{a}\), written as an exponent. Calculators and programming languages use this form. |
| \(b'\) | b prime | The "new estimate" in the repeat-by-hand method. The prime mark shows that it is \(b\) improved by one step. |
Terms
| power (exponentiation) | Multiplying the same number by itself several times. In \(2^3 = 2 \times 2 \times 2 = 8\), the small number at the upper right (the exponent) tells how many times. |
| nth root (radical) | The number that gives \(a\) when raised to the power \(n\) (\(\sqrt[n]{a}\)). It is the general name for square roots, cube roots and higher roots, and it is exactly the reverse of raising to a power. |
| square root | A number whose square is \(a\). A positive number has two square roots, \(+\sqrt{a}\) and \(-\sqrt{a}\), and the radical sign \(\sqrt{a}\) means the positive one. Taught in Grade 8. |
| cube root | A number that gives \(a\) when cubed (\(\sqrt[3]{a}\)). It comes up when you find the edge length of a cube from its volume. |
| index | The number \(n\) that tells which root it is. It is written small at the upper left of the radical sign. Index 2 is the square root and index 3 is the cube root. |
| exponent | The small number at the upper right of a number that tells how many times to multiply. If we allow the fraction \(\frac{1}{n}\) as an exponent, a root can be written as a power (\(a^{1/n}\)). |
| radicand | The number under the radical sign, \(a\) in \(\sqrt[n]{a}\). On this page it is called "the number under the root". |
| principal square root | The square root that is 0 or positive. The radical sign \(\sqrt{a}\) always means the principal square root, so \(\sqrt{9} = 3\), not \(-3\). |
| irrational number | A number that cannot be written as a fraction, like \(\sqrt{2} = 1.41421356\ldots\) (its decimal goes on forever without repeating). A root that does not come out exactly is an irrational number. |
| Newton's method | A method that gets closer and closer to the answer of an equation by improving an estimate with a formula, again and again. It is the basis of how computers quickly find square roots and other roots. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
| Powers and exponents (Grade 6 and up) |
|
| Square roots (Grade 8) |
|
| Multiplying negative numbers (Grade 7) |
|
| Multiplying decimals (Grade 5) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the square root (√a)
| Number under the root a | 27 |
| Square root √a | =SQRT(B1) |
Table to find the cube root
| Number under the root a | 27 |
| Cube root | =B1^(1/3) |
Table to find the nth root
| Index n (which root) | 8 |
| Number under the root a | 15 |
| nth root | =B2^(1/B1) |
The first table shows the square root of 27 (about 5.196) in B2, the second shows the cube root of 27 (3) in B2, and the third shows the 8th root of 15 (about 1.403) in B3.
SQRT is a function for square roots only. For cube roots and other roots, use "^(1/n)" (the power 1/n). "^" is the symbol for raising to a power.
Note: If the number under the root is negative, Excel gives an error (#NUM!) even with "^(1/3)". For an odd root of a negative number, take the sign off first, as in "=-((-B1)^(1/3))". (Excel applies the minus sign before the power, so if you leave out the outer parentheses you still get the error.)
SQRT is a function for square roots only. For cube roots and other roots, use "^(1/n)" (the power 1/n). "^" is the symbol for raising to a power.
Note: If the number under the root is negative, Excel gives an error (#NUM!) even with "^(1/3)". For an odd root of a negative number, take the sign off first, as in "=-((-B1)^(1/3))". (Excel applies the minus sign before the power, so if you leave out the outer parentheses you still get the error.)
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the square root (√a)
| Number under the root a | 27 |
| Square root √a | =SQRT(B1) |
Table to find the cube root
| Number under the root a | 27 |
| Cube root | =B1^(1/3) |
Table to find the nth root
| Index n (which root) | 8 |
| Number under the root a | 15 |
| nth root | =B2^(1/B1) |
The same formulas as in Excel work as is. Copy the whole table, paste it into cell A1, and replace the input numbers with your own.
Google Sheets also gives an error if the number under the root is negative. For an odd root of a negative number, take the sign off first, as in "=-((-B1)^(1/3))". (The minus sign is applied before the power, so if you leave out the outer parentheses you still get the error.)
Google Sheets also gives an error if the number under the root is negative. For an odd root of a negative number, take the sign off first, as in "=-((-B1)^(1/3))". (The minus sign is applied before the power, so if you leave out the outer parentheses you still get the error.)
How to calculate it in Python
number = 27 # number under the root, a
degree = 3 # which root (index n)
root = number ** (1 / degree) # nth root = a to the power (1/n)
print(f"Root (index {degree}) of {number}: {root}")
# For an odd root of a negative number, take the sign off first
# (raising a negative number to the power (1/3) directly makes Python return a complex number)
negative_number = -8
odd_degree = 3
negative_root = -((-negative_number) ** (1 / odd_degree))
print(f"Root (index {odd_degree}) of {negative_number}: {negative_root}")
Runs with the standard library only. "**" is the symbol for raising to a power, and the power (1/n) is the nth root. Change the number and the index at the top and run it. For an odd root of a negative number, take the sign off first, as in the example.
How to write it in LaTeX and other math languages (copy and paste)
Definition of the \(n\)th root (the number that gives \(a\) when raised to the power \(n\))
ⁿ√a = b ⟺ bⁿ = a
\sqrt[n]{a} = b \iff b^{n} = a
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mroot><mi>a</mi><mi>n</mi></mroot>
<mo>=</mo>
<mi>b</mi>
<mo>⟺</mo>
<msup><mi>b</mi><mi>n</mi></msup>
<mo>=</mo>
<mi>a</mi>
</mrow>
</math>
root(n)(a) = b iff b^n = a
Surd[a, n]
b := root(a, n);
b = nthroot(a, n);
√(n&a) = b ⟺ b^n = a
Writing a root as an exponent (\(\sqrt[n]{a} = a^{1/n}\))
ⁿ√a = a^(1/n)
\sqrt[n]{a} = a^{1/n}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mroot><mi>a</mi><mi>n</mi></mroot>
<mo>=</mo>
<msup>
<mi>a</mi>
<mrow><mn>1</mn><mo>/</mo><mi>n</mi></mrow>
</msup>
</mrow>
</math>
root(n)(a) = a^(1/n)
a^(1/n)
b := a^(1/n);
b = a^(1/n);
√(n&a) = a^(1/n)
Roots of negative numbers (only when the index is odd)
ⁿ√(−a) = −(ⁿ√a)
\sqrt[n]{-a} = -\sqrt[n]{a}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mroot>
<mrow><mo>−</mo><mi>a</mi></mrow>
<mi>n</mi>
</mroot>
<mo>=</mo>
<mo>−</mo>
<mroot><mi>a</mi><mi>n</mi></mroot>
</mrow>
</math>
root(n)(-a) = -root(n)(a)
Surd[-a, n]
b := surd(-a, n);
b = nthroot(-a, n);
√(n&−a) = −√(n&a)
Finding a square root by hand (averaging again and again)
b′ = (b + a ÷ b) ÷ 2
b' = \dfrac{1}{2}\left( b + \dfrac{a}{b} \right)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msup><mi>b</mi><mo>′</mo></msup>
<mo>=</mo>
<mfrac>
<mrow>
<mi>b</mi>
<mo>+</mo>
<mfrac><mi>a</mi><mi>b</mi></mfrac>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</math>
b' = (b + a/b)/2
(b + a/b)/2
bnew := (b + a/b)/2;
b_new = (b + a/b)/2;
b′ = (b + a/b)/2
How to have ChatGPT do the calculation
You are a calculation assistant for numbers. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). Find each of the following roots. 1. The square root of 27 2. The cube root of 10 3. The 8th root of 15 4. The cube root of −8 (for an odd root of a negative number, take the sign off first) Show the formulas you used and the numbers from the execution result.
How to Use
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1Enter your numbersType the numbers you want to calculate with into the input fields
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2CalculatePress the "Calculate" button
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3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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