## Cubes and Cube Roots

3.2  Cubes and Cube Roots

Cubes:

 Definition A number is multiplied by the number itself three times.

• Examples: $$3^3,\,7^3,\,12^3$$

Perfect cubes:

 Definition A number equal to the cube of a whole number.

• Examples: $$1,\,8,\,27$$

Determine a number is a perfect cube:

• Perfect cube can be written as a product of three equal factors.

 Example \begin{aligned} 125&=5\times5\times5 \\\\&=5^3. \end{aligned} $$125$$ is a perfect cube.

Relationship between cubes and cube roots:

• Finding the cube and finding the cube root are inverse operations.

 Example The cube of $$8$$ is $$512$$. The cube root of $$512$$ is $$8$$. $$8\times8\times8=512$$ Thus, \begin{aligned} \sqrt[3]{512}&=\sqrt[3]{8\times8\times8} \\\\&=8. \end{aligned}

The cube of a number:

 Example Calculate: (i) \begin{aligned} 7^3&=7\times7\times7 \\\\&=343. \end{aligned} (ii) \begin{aligned}&\space \bigg(-\dfrac{1}{4}\bigg)^3\\\\&=\bigg(-\dfrac{1}{4}\bigg)\times\bigg(-\dfrac{1}{4}\bigg)\times\bigg(-\dfrac{1}{4}\bigg) \\\\&=-\dfrac{1}{64}. \end{aligned}

The cube root of a number:

 Example Solve: (i) \begin{aligned} \sqrt[3]{1\,000}&=\sqrt[3]{10\times10\times10} \\\\&=\sqrt[3]{10^3}\\\\&=10. \end{aligned} (ii) \begin{aligned} \sqrt[3]{\dfrac{27}{64}}&=\sqrt[3]{\dfrac{3}{4}\times\dfrac{3}{4}\times\dfrac{3}{4}} \\\\&=\sqrt[3]{\bigg(\dfrac{3}{4}\bigg)^3} \\\\&=\dfrac{3}{4}. \end{aligned}

Computation involving different operations on squares, square roots, cubes and cube roots:

1. Find the value of squares, square roots, cubes or cube roots.
2. Solve the operation in the brackets.
3. Solve the operations $$\times$$ and $$\div$$ from left to right.
4. Solve the operations $$+$$ and $$-$$ from left to right.

 Example Calculate: (i) \begin{aligned} &\space\sqrt[3]{64}+0.2^2\\\\&=4+0.04 \\\\&=4.04. \end{aligned} (ii) \begin{aligned} &\space\sqrt[3]{-64}\times(5^3+0.1^2)\\\\&=-4\times(125+0.01)\\\\&=-4\times125.01 \\\\&=-500.04. \end{aligned}