| 3.2 |
Cubes and Cube Roots |
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| Cubes: |
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Definition |
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A number is multiplied by the number itself three times.
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- Examples: \(3^3,\,7^3,\,12^3\)
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| Perfect cubes: |
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Definition |
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A number equal to the cube of a whole number.
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| Determine a number is a perfect cube: |
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- Perfect cube can be written as a product of three equal factors.
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Example |
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\(\begin{aligned} 125&=5\times5\times5 \\\\&=5^3. \end{aligned}\)
\(125\) is a perfect cube.
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| Relationship between cubes and cube roots: |
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- Finding the cube and finding the cube root are inverse operations.
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Example |
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The cube of \(8\) is \(512\).
The cube root of \(512\) is \(8\).
\(8\times8\times8=512\)
Thus,
\(\begin{aligned} \sqrt[3]{512}&=\sqrt[3]{8\times8\times8} \\\\&=8. \end{aligned}\)
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| The cube of a number: |
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Example |
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Calculate:
(i)
\(\begin{aligned} 7^3&=7\times7\times7 \\\\&=343. \end{aligned}\)
(ii)
\(\begin{aligned}&\space \bigg(-\dfrac{1}{4}\bigg)^3\\\\&=\bigg(-\dfrac{1}{4}\bigg)\times\bigg(-\dfrac{1}{4}\bigg)\times\bigg(-\dfrac{1}{4}\bigg) \\\\&=-\dfrac{1}{64}. \end{aligned}\)
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| The cube root of a number: |
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Example |
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Solve:
(i)
\(\begin{aligned} \sqrt[3]{1\,000}&=\sqrt[3]{10\times10\times10} \\\\&=\sqrt[3]{10^3}\\\\&=10. \end{aligned}\)
(ii)
\(\begin{aligned} \sqrt[3]{\dfrac{27}{64}}&=\sqrt[3]{\dfrac{3}{4}\times\dfrac{3}{4}\times\dfrac{3}{4}} \\\\&=\sqrt[3]{\bigg(\dfrac{3}{4}\bigg)^3} \\\\&=\dfrac{3}{4}. \end{aligned}\)
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| Computation involving different operations on squares, square roots, cubes and cube roots: |
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- 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.
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Example |
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Calculate:
(i)
\(\begin{aligned} &\space\sqrt[3]{64}+0.2^2\\\\&=4+0.04 \\\\&=4.04. \end{aligned}\)
(ii)
\(\begin{aligned} &\space\sqrt[3]{-64}\times(5^3+0.1^2)\\\\&=-4\times(125+0.01)\\\\&=-4\times125.01 \\\\&=-500.04. \end{aligned}\)
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