1.04 Cube roots of perfect cubes
Introduction
We have learned how to find the square root of perfect squares .
We are now ready to learn about perfect cubes and finding cube roots. We have recognized that squaring a number and taking the square root of a number are inverse operations, likewise, cubing a number and taking the cube root are inverse operations.
Cube roots of perfect cubes
Raising a number to the power of three is often called "cubing" a number. For example, the expression x^3 has the following meanings:
- x^3
A number x raised to the power of three
x cubed
Let's look at a table of the first ten perfect cubes. A perfect cube is a number that can be expressed as the cube of an integer. For example, the number 8 is a perfect cube because it can be expressed as 2 \times 2 \times 2 or 2^3.
| \enspace \, 1 | = | \enspace \, 1 \times 1 \times 1 | = | 1^3 |
| \enspace \, 8 | = | \enspace \, 2 \times 2 \times 2 | = | 2^3 |
| \enspace 27 | = | \enspace \, 3 \times 3 \times 3 | = | 3^3 |
| \enspace 64 | = | \enspace \, 4 \times 4 \times 4 | = | 4^3 |
| \, 125 | = | \enspace \, 5 \times 5 \times 5 | = | 5^3 |
| \, 216 | = | \enspace \, 6 \times 6 \times 6 | = | 6^3 |
| \, 343 | = | \enspace \, 7 \times 7 \times 7 | = | 7^3 |
| \, 512 | = | \enspace \, 8 \times 8 \times 8 | = | 8^3 |
| \, 729 | = | \enspace \, 9 \times 9 \times 9 | = | 9^3 |
| 1000 | = | 10 \times 10 \times 10 | = | 10^3 |
If we are asked to find the cube root of a value, we are being asked, "What number multiplied by itself three times would give this value?"
We might also see the cube root symbol written with a number inside it, for example, \sqrt[3]{125} represents the cube root of 125 which is equivalent to 5 because 5 \times 5 \times 5 = 125.
We can find the cube root of a perfect cube by thinking of a number that, when multiplied to itself three times, will equal the number underneath the cube root symbol. It will help to be familiar with the first ten perfect cubes.
Another way to find the cube root is using the prime factorization method. We can follow these steps:
Begin with the prime factorization of the given number.
Partition the factors into groups of the same three factors.
Remove the cube root symbol and multiply the factors to get the answer.
Let's look at the following worked questions to apply these steps.
Examples
Example 1
Evaluate \sqrt[3]{27}.
Example 2
Solve x^3 = 64.
Example 3
What is \sqrt[3] {216} ?
A perfect cube is a number that can be expressed as the cube of an integer. \begin{aligned} 125 &= 5\times 5 \times 5 \\ &=5^3 \end{aligned}
If we are asked to find the cube root of a value, we are being asked, "What number multiplied by itself three times would give this value?" \begin{aligned} \sqrt[3]{125} &= \sqrt[3]{5\times 5 \times 5} \\ &=5 \end{aligned}
Another way to find the cube root is using prime factorization method. We can follow these steps:
Begin with the prime factorization of the given number.
Partition the factors into groups of three same factors.
Remove the cube root symbol and multiply the factors to get the answer.