What a cube root is and why you need it
A cube root is the number that, when multiplied by itself three times, gives you your starting number. If you cube 3 (meaning 3 × 3 × 3), you get 27. The cube root of 27 is 3. You will see cube root written as ∛27 or as 27^(1/3) in formulas and spreadsheets.
You need cube roots when you are working with volume, density, or any measurement where three dimensions multiply together. If you know a cube's volume and need to find the length of one side, you take the cube root of that volume. The same applies to scaling recipes, calculating storage space, or solving physics problems.
You have three practical ways to find a cube root: use a calculator, use a spreadsheet formula, or estimate it by hand. Which one you choose depends on whether you need an exact answer, how much precision matters, and whether you have tools nearby.
Key Takeaways
- A cube root is the number that multiplies by itself three times to reach your starting number; ∛27 = 3 because 3 × 3 × 3 = 27.
- On a scientific calculator, enter the number, press the cube root button (often labeled ∛x or marked as a function), and read the result.
- In Excel or Google Sheets, type =POWER(number, 1/3) to calculate any cube root when ready.
- By hand, you can estimate by testing whole numbers or use the Newton-Raphson method for closer guesses with each step.
- Negative numbers have cube roots too: the cube root of −8 is −2, because −2 × −2 × −2 = −8.
Using a scientific calculator
Most scientific calculators have a dedicated cube root button. Look for a button labeled ∛x, or sometimes x^(1/3), or a menu option under "Functions" or "Math". The exact location varies by brand — Texas Instruments, Casio, and HP all place it differently.
To find the cube root of 125: enter 125, press the cube root button, and the display shows 5. That is all. If your calculator does not have a dedicated button, you can use the power function instead. Enter 125, press the exponent or power button (usually marked ^ or x^y), then enter 0.333 (or 1/3 if your calculator accepts fractions), and press equals. The result is the same.
Calculators give you the answer when ready and accurately. This is the fastest method when you have a calculator in front of you and precision matters.
Computing cube root in Excel or Google Sheets
Spreadsheets handle cube roots through the POWER function. In either Excel or Google Sheets, type this formula into any cell: =POWER(number, 1/3). Replace "number" with the actual value or a cell reference.
Example: to find the cube root of 64, type =POWER(64, 1/3) and press Enter. The cell displays 4. If your number is in cell A1, type =POWER(A1, 1/3) instead. You can also use the caret symbol: =A1^(1/3) works the same way in most spreadsheets.
This method is useful when you are working with a list of numbers and need to calculate the cube root for each one. You can copy the formula down a column and explore it to hundreds of values at once. Spreadsheets also let you round the result if you want fewer decimal places: =ROUND(POWER(64, 1/3), 2) rounds to two decimal places.
Estimating cube root by testing whole numbers
If you do not have a calculator or spreadsheet, you can narrow down the answer by testing. Start by cubing small whole numbers and building a mental reference: 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 6³ = 216, 7³ = 343, 8³ = 512, 9³ = 729, 10³ = 1000.
To estimate the cube root of 300: you know that 6³ = 216 and 7³ = 343, so the cube root of 300 is between 6 and 7, closer to 7. You can narrow it further by testing 6.7: 6.7 × 6.7 × 6.7 = 300.763, which is very close. This method takes longer but works when you need only a rough answer or want to check whether a calculator result makes sense.
The more reference cubes you memorize, the faster you can estimate. For most everyday problems, knowing the cubes of 1 through 10 is enough to get within one decimal place of the true answer.
Using the Newton-Raphson method for closer estimates
The Newton-Raphson method is a mathematical technique that gets you closer to the true cube root with each step. It requires a starting guess and a straightforward calculation repeated as many times as you want precision.
The formula is: new guess = (2 × old guess + number ÷ (old guess²)) ÷ 3. Start with any reasonable guess — if you are finding the cube root of 100, guess 4 or 5. Then plug that guess into the formula, calculate the result, and use that result as your next guess. Repeat two or three times and you will have a very accurate answer.
Example: to find the cube root of 100, start with a guess of 4. Calculate: (2 × 4 + 100 ÷ (4²)) ÷ 3 = (8 + 100 ÷ 16) ÷ 3 = (8 + 6.25) ÷ 3 = 4.75. Now guess 4.75 and repeat: (2 × 4.75 + 100 ÷ (4.75²)) ÷ 3 = 4.642. One more time: (2 × 4.642 + 100 ÷ (4.642²)) ÷ 3 = 4.6416. The true cube root of 100 is 4.6416, so you have arrived at the answer in three steps.
This method is slower than a calculator but faster than random guessing, and it works on paper or in your head. It is useful when you want to understand how the answer was found or when you are teaching someone else.
Cube roots of negative numbers
Negative numbers have cube roots, and the rule is straightforward: the cube root of a negative number is negative. The cube root of −8 is −2, because −2 × −2 × −2 = −8. The cube root of −27 is −3.
Calculators and spreadsheets handle this correctly. If you are estimating by hand, remember that cubing a negative number gives a negative result, so work with the positive version of your number, find its cube root, and then add the negative sign.
This is different from square roots, where negative numbers have no real answer. Cube roots always exist for any real number, positive or negative.
When to use each method
Use a calculator when you need a fast, accurate answer and you have one available. Use a spreadsheet when you are working with multiple numbers or building a model where the cube root might change. Use hand estimation when you are checking whether an answer is reasonable, teaching someone the concept, or working without tools. Use Newton-Raphson when you want precision without a calculator and do not mind the extra steps.
For most people most of the time, a calculator or spreadsheet is the practical choice. But knowing how to estimate by hand means you can solve the problem anywhere, and understanding the Newton-Raphson method shows you how mathematicians actually solve these problems when they cannot rely on a button.
Frequently Asked Questions
What is the difference between a cube root and a square root?
A square root is a number multiplied by itself twice (two dimensions). A cube root is a number multiplied by itself three times (three dimensions). The square root of 9 is 3 because 3 × 3 = 9. The cube root of 27 is 3 because 3 × 3 × 3 = 27. Square roots of negative numbers do not exist in real numbers, but cube roots of negative numbers do.
Can I use a regular calculator or do I need a scientific one?
A scientific calculator has a cube root button and makes the job when ready. A basic four-function calculator does not have this button. However, you can still find cube roots on a basic calculator by using the power function if it has one, or by using the estimation method described above.
Why does the spreadsheet formula use 1/3 as the exponent?
Because raising a number to the power of 1/3 is mathematically the same as taking its cube root. This works because (x^(1/3))³ = x. Spreadsheets do not have a separate cube root function like they do for square roots, so the power function with an exponent of 1/3 is the standard way to calculate it.
How accurate do I need to be?
That depends on what you are using the cube root for. If you are calculating the side length of a storage box, one or two decimal places is usually enough. If you are solving a physics equation or engineering problem, you may need three or more decimal places. A calculator gives you as many decimal places as the display allows, which is more than enough for most purposes.
What if the cube root is not a whole number?
Most cube roots are not whole numbers. The cube root of 10 is about 2.154, the cube root of 50 is about 3.684, and so on. Calculators and spreadsheets show you the decimal answer. If you are estimating by hand, you can round to the nearest tenth or hundredth depending on how precise you need to be.