6.3 Cube Roots

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. It is denoted as $ \sqrt[3]{a} $ or $ a^{1/3} $, where $a$ is the number whose cube root we are finding.

For example, the cube root of 8 is 2 because $ 2 \times 2 \times 2 = 8 $. Thus, we write:

$$ \sqrt[3]{8} = 2 $$

Cube roots can also be negative. For example, the cube root of -8 is -2 because $ (-2) \times (-2) \times (-2) = -8 $. Hence:

$$ \sqrt[3]{-8} = -2 $$

Properties of Cube Roots

• The cube root of 0 is 0: $$ \sqrt[3]{0} = 0 $$
• Cube roots are real numbers for both positive and negative numbers.
• Cube root of a positive number is positive, and the cube root of a negative number is negative.
• Cube roots are also distributive over multiplication: $$ \sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b} $$
• Similarly, cube roots are distributive over division: $$ \sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}} $$

Cube Root of Perfect Cubes

A perfect cube is a number that is the cube of an integer. For example, 8 is a perfect cube because $ 2^3 = 8 $, and 27 is a perfect cube because $ 3^3 = 27 $. The cube root of a perfect cube is simply the integer that was cubed to get the original number. Some examples of perfect cubes and their cube roots are:

• $$ \sqrt[3]{1} = 1 $$
• $$ \sqrt[3]{8} = 2 $$
• $$ \sqrt[3]{27} = 3 $$
• $$ \sqrt[3]{64} = 4 $$
• $$ \sqrt[3]{125} = 5 $$

Estimating Cube Roots

For numbers that are not perfect cubes, we can estimate the cube root. For example, to estimate $ \sqrt[3]{20} $, we note that $ 2^3 = 8 $ and $ 3^3 = 27 $. Since 20 is between 8 and 27, we know that $ \sqrt[3]{20} $ is between 2 and 3. By trial and error or using a calculator, we can get a more accurate value:

$$ \sqrt[3]{20} \approx 2.728 $$

Cube Root Function

The cube root function, like other root functions, is a continuous function that is defined for all real numbers. Its graph passes through the origin (0,0) and has a curve similar to the square root function but is less steep. The cube root function is an increasing function, meaning that as the input number increases, the cube root also increases.

Applications of Cube Roots

Cube roots are used in various real-life applications, including:

• In geometry, to find the side length of a cube given its volume. If the volume of a cube is $ V $, the side length $ s $ is given by $ s = \sqrt[3]{V} $. For example, if the volume of a cube is 27 cubic units, the side length is $ \sqrt[3]{27} = 3 $ units.
• In physics, for calculating certain physical properties like density, pressure, and others, where cube roots might be involved.