Cube Roots
A cube root of a number is the value you multiply by itself three times to get that number. The cube root of 27 is 3, because 3 \times 3 \times 3 = 27. We write it as \sqrt[3]{27} = 3. Unlike square roots, negative numbers have cube roots, so \sqrt[3]{-8} = -2.
Before this lesson: You should know cubes from Exponents and the ideas in Square Root. You will also use multiplying integers for negative numbers.
What Is a Cube Root?
Cubing a number means using it as a factor three times: 3^3 = 3 \times 3 \times 3 = 27. A cube root goes the other way. It asks, “What number, used three times, gives 27?”
Picture a big cube built from small cubes. If it holds 27 small cubes, each edge is 3 cubes long. The volume is 27, and the cube root of the volume is the edge length.
The Cube Root Rule and Perfect Cubes
\sqrt[3]{a} = b \quad \text{means} \quad b \times b \times b = aThe small 3 in \sqrt[3]{\phantom{a}} is called the index. It tells you this is a cube root, not a square root. The number inside is the radicand, just like with square roots.
A perfect cube is a number made by cubing a whole number. These ten are worth knowing by heart.
| Cube | Cube root | Cube | Cube root |
|---|---|---|---|
| 1^3 = 1 | \sqrt[3]{1} = 1 | 6^3 = 216 | \sqrt[3]{216} = 6 |
| 2^3 = 8 | \sqrt[3]{8} = 2 | 7^3 = 343 | \sqrt[3]{343} = 7 |
| 3^3 = 27 | \sqrt[3]{27} = 3 | 8^3 = 512 | \sqrt[3]{512} = 8 |
| 4^3 = 64 | \sqrt[3]{64} = 4 | 9^3 = 729 | \sqrt[3]{729} = 9 |
| 5^3 = 125 | \sqrt[3]{125} = 5 | 10^3 = 1000 | \sqrt[3]{1000} = 10 |
Cube Roots of Negative Numbers
A negative number times itself three times stays negative. For example, (-2) \times (-2) \times (-2) = -8. So \sqrt[3]{-8} = -2.
This is a big difference from square roots. A negative number has no square root you know, but it always has a cube root. The cube root of a negative number is negative.
Cube Roots of Fractions and Decimals
For a fraction, take the cube root of the top and the bottom.
\sqrt[3]{\frac{8}{27}} = \frac{\sqrt[3]{8}}{\sqrt[3]{27}} = \frac{2}{3}For a decimal, find the decimal that cubes to the number. For example, \sqrt[3]{0.008} = 0.2 because 0.2 \times 0.2 \times 0.2 = 0.008. The answer has one third as many decimal places as the radicand.
How to Estimate a Cube Root
For a number that is not a perfect cube, find the perfect cubes just below and just above it. The cube root is between their cube roots.
For \sqrt[3]{40}, the nearby perfect cubes are 27 and 64. So \sqrt[3]{40} is between 3 and 4. Since 40 is closer to 27, the answer is closer to 3. A calculator gives \sqrt[3]{40} \approx 3.42.
Worked Examples
Example 1: Cube root of a perfect cube
Find \sqrt[3]{64}.
Step 1. Ask what number, used three times, gives 64.
Step 2. Try 4: 4 \times 4 = 16, and 16 \times 4 = 64.
Answer: \sqrt[3]{64} = 4
Example 2: Cube root of a negative number
Find \sqrt[3]{-125}.
Step 1. Ignore the sign first. Since 5^3 = 125, the number is 5 or −5.
Step 2. The radicand is negative, so the cube root is negative. Check: (-5) \times (-5) \times (-5) = -125.
Answer: \sqrt[3]{-125} = -5
Example 3: Cube root of a fraction and a decimal
Find \sqrt[3]{\frac{27}{64}} and \sqrt[3]{0.064}.
Step 1. For the fraction, \sqrt[3]{27} = 3 and \sqrt[3]{64} = 4. So the answer is \frac{3}{4}.
Step 2. For the decimal, \sqrt[3]{64} = 4, so the digit is 4. The radicand has 3 decimal places, so the answer has 1. Try 0.4.
Step 3. Check: 0.4 \times 0.4 \times 0.4 = 0.064.
Answer: \sqrt[3]{\frac{27}{64}} = \frac{3}{4} and \sqrt[3]{0.064} = 0.4
Example 4: Estimate a cube root
Estimate \sqrt[3]{100} to one decimal place.
Step 1. The perfect cubes around 100 are 64 and 125. So \sqrt[3]{100} is between 4 and 5.
Step 2. 100 is 36 away from 64 and 25 away from 125. It is a bit closer to 125, so the answer is a bit more than 4.5.
Step 3. Test 4.6: 4.6^3 \approx 97.3. Test 4.7: 4.7^3 \approx 103.8. 97.3 is closer to 100.
Answer: \sqrt[3]{100} \approx 4.6
Example 5: Word problem with a cube-shaped box
A box shaped like a cube holds 216 cubic inches. How long is each edge? What is the area of one face?
Step 1. The volume of a cube is edge × edge × edge. So the edge is \sqrt[3]{216}.
Step 2. 6 \times 6 \times 6 = 216, so each edge is 6 inches.
Step 3. Each face is a square. Its area is 6 \times 6 = 36 square inches.
Answer: Each edge is 6 inches, and one face has an area of 36 square inches.
Common Mistakes with Cube Roots
| Mistake | Example of the error | Fix |
|---|---|---|
| Dividing by 3 | \sqrt[3]{27} = 9 | Ask what number, used three times, gives 27. 3 \times 3 \times 3 = 27, so the answer is 3. |
| Using only two factors | \sqrt[3]{9} = 3 | 3 \times 3 = 9 is squaring. For a cube root, check 3 \times 3 \times 3 = 27. |
| Saying a negative has no cube root | “\sqrt[3]{-27} has no answer” | Three negative factors give a negative. \sqrt[3]{-27} = -3. |
| Mixing up the index | Reading \sqrt[3]{8} as 3 \times \sqrt{8} | The small 3 is the index. It is part of the root sign, not a factor. |
Square Roots vs. Cube Roots
| Question | Square root | Cube root |
|---|---|---|
| Symbol | \sqrt{a} | \sqrt[3]{a} |
| It undoes | squaring | cubing |
| Number of equal factors | 2 | 3 |
| Shape | side of a square from its area | edge of a cube from its volume |
| Root of a negative number | none in the numbers you know | a negative number |
Square roots and cube roots both go with exponents. In Algebra 1, you will see that roots can also be written as exponents.
Practice Problems
- Find \sqrt[3]{343}.
- Find \sqrt[3]{729}.
- Find \sqrt[3]{1000}.
- Find \sqrt[3]{-1}.
- Find \sqrt[3]{-216}.
- Find \sqrt[3]{\frac{1}{125}}.
- Find \sqrt[3]{0.027}.
- Between which two whole numbers is \sqrt[3]{30}? Which one is it closer to?
- A cube-shaped tank holds 512 cubic feet of water. How long is each edge?
Show answers
- 7, because 7 \times 7 \times 7 = 343.
- 9, because 9 \times 9 \times 9 = 729.
- 10, because 10 \times 10 \times 10 = 1000.
- −1, because (-1) \times (-1) \times (-1) = -1.
- −6, because (-6) \times (-6) \times (-6) = -216.
- \frac{1}{5}, because \sqrt[3]{1} = 1 and \sqrt[3]{125} = 5.
- 0.3, because 0.3 \times 0.3 \times 0.3 = 0.027.
- 30 is between 27 and 64, so \sqrt[3]{30} is between 3 and 4. It is much closer to 3, because 30 is only 3 away from 27. (It is about 3.1.)
- \sqrt[3]{512} = 8, because 8 \times 8 \times 8 = 512. Each edge is 8 feet.
Cube Root FAQ
What is the cube root of 8?
The cube root of 8 is 2, because 2 \times 2 \times 2 = 8.
Can you take the cube root of a negative number?
Yes. Three negative factors give a negative answer, so every negative number has a negative cube root. For example, \sqrt[3]{-64} = -4.
What is the cube root of 2?
The cube root of 2 is about 1.26. It is not a whole number, because 2 is between the perfect cubes 1 and 8.
Is a cube root the same as dividing by 3?
No. Dividing 27 by 3 gives 9, but 9 \times 9 \times 9 = 729, not 27. The cube root of 27 is 3, because 3 \times 3 \times 3 = 27.
Summary
- A cube root of a number, used as a factor three times, gives the number: \sqrt[3]{27} = 3.
- The small 3 in the root sign is the index.
- Perfect cubes like 1, 8, 27, 64 and 125 have whole number cube roots.
- The cube root of a negative number is negative: \sqrt[3]{-8} = -2.
- For a fraction, take the cube root of the top and the bottom.
- To estimate, find the perfect cubes on each side of the number.
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