Cubes and cube roots

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:

Find the value of squares, square roots, cubes or cube roots.
Solve the operation in the brackets.
Solve the operations \(\times\) and \(\div\) from left to right.
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)

\(\begin{aligned} &\space\sqrt[3]{-64}\times(5^3+0.1^2)\\\\&=-4\times(125+0.01)\\\\&=-4\times125.01 \\\\&=-500.04. \end{aligned}\)