The Basics Of Cube Roots (Examples & Answers)

The cube root gets its name from geometry. A cube is a three-dimensional figure with equal sides, and each side is the cube root of the volume. To see why this is true, consider how you determine the volume (​V​) of a cube. You multiply the length by the width and also by the depth. Since all three are equal, this is equivalent to multiplying the length of one side (​l​) by itself twice: Volume = (​l​ × ​l​ × ​l​) = ​l​³. If you know the volume of the cube, the length of each side is therefore the cube root of the volume:

\(l = \sqrt[3]{V}\)

In other words, the cube root of one number is a second number which, when multiplied by itself twice, produces the original number. Mathematicians represent cube root with a radical sign preceded by a superscript 3.

Scientific calculators usually include a function that automatically displays the cube root of any number, and it's a good thing, because finding the cube root of a random number usually isn't easy. However, if the cube root is a non-fractional integer between 1 and 100, a simple trick makes it easy to find. For this trick to work, though, you need to cube the integers from 1 to 10, make a table and memorize the values.

Multiply 1 by itself twice and the answer is still 1, so the cube root of 1 is 1. Multiply 2 by itself twice, and the answer is 8, so the cube root of 8 is 2. Similarly, the cube root of 27 is 3, the cube root of 64 is 4 and the cube root of 125 is 5. You can continue this procedure from 6 to 10 to find

\(\sqrt[3]{216}=6\
\sqrt[3]{343}=7 \
\sqrt[3]{512}=8 \
\sqrt[3]{729}=9 \
\sqrt[3]{1000}=10\)

Once you have memorized these values, the rest of the procedure is straightforward. The last digit of the original number corresponds to the last digit of the number you're looking for, and you find the first digit of the cube root by looking at the first three digits in the original number.

In general, the most reliable method for finding the cube root of a random number is trial and error. Make your best guess, cube that number, and see how close it is to the number for which you're trying to find the cube root, then refine your guess.

For example, you know ³√3 has to be between 1 and 2, because 1³ = 1 and 2³ = 8. Try multiplying 1.5 by itself twice, and you get 3.375. That's too high. If you multiply 1.4 by itself twice, you get 2.744, which is too low. It turns out ³√3 is an irrational number, and accurate to six decimal places, it is 1.442249. Because it's irrational, no amount of trial and error will produce a completely accurate result. Be thankful for your calculator!

You can often simplify larger numbers by factoring out smaller numbers. This is the case when finding the cube root of 81. You can divide 81 by 3 to get 27, then divide by 3 again to get 9, and divide once more by 3 to get 3. In this way:

\(\sqrt[3]{81} =\sqrt[3]{3 × 3 × 3 × 3}\)

Remove the first three 3's from the radical sign, and you're left with

\(\sqrt[3]{81} = 3 \sqrt[3]{3}\)

\(\sqrt[3]{3} = 1.442249\)
\(\text{so }\sqrt[3]{81} = 3 × 1.442249 = 4.326747\)

which is also an irrational number.

​1. What is​

\(\sqrt[3]{150} = ?\)

Note that

\(\sqrt[3]{125} = 5 \text{ and } \sqrt[3]{216} = 6\)

so the number you're looking for is between 5 and 6, and closer to 5 than 6. (5.4)³ = 157.46, which is too high, and (5.3)³ is 148.88, which is slightly too low. (5.35)³ = 153.13 is too high. (5.31)³ = 149.72 is too low. Continuing this process, you find the correct value, accurate to six decimal places: 5.313293.

​2. What is​

\(\sqrt[3]{1,029}=?\)

It's always a good idea to look for factors in large numbers. In this case, it turns out 1029 ÷ 7 = 147; 147 ÷ 7 = 21 and 21 ÷ 7 = 3. We can therefore rewrite 1,029 as (7 × 7 × 7 × 3), and we get:

\(\sqrt[3]{1029}=7\sqrt[3]{3} = 10.095743\)

​3. What is​

\(\sqrt[3]{-27}\)

Unlike square roots of negative numbers, which are imaginary, cube roots are simply negative. In the case, the answer is −3.