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| 3.1 |
Squares and Square Roots |
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| Squares: |
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Definition |
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A number that indicates a number is multiplied by the number itself.
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- Examples: \(2^2,\, 9^2,\, 5^2\)
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| Perfect squares: |
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Defintion |
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A number that is equal to the square of a whole number.
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- Examples: \(1,\, 4,\, 9\)
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| Determine a number is a perfect square: |
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- Perfect square can be written as a product of two equal factors.
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Example |
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\(\begin{aligned} 144&=12\times12 \\\\&=12^2. \end{aligned}\)
\(144\) is a perfect square.
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| Relationship between squares and square roots: |
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- Finding the square and finding the square root are inverse operations.
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Example |
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The square of \(8\) is \(64\).
The square root of \(64\) is \(8\).
\(8\times8=64\)
Thus,
\(\begin{aligned} \sqrt{64}&=\sqrt{8\times8} \\\\&=8. \end{aligned}\)
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| The square of a number: |
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Example |
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Calculate:
(i)
\(\begin{aligned} 3^2&=3\times3 \\\\&=9. \end{aligned}\)
(ii)
\(\begin{aligned} \bigg(\dfrac{2}{5}\bigg)^2&=\dfrac{2}{5}\times\dfrac{2}{5} \\\\&=\dfrac{4}{25}. \end{aligned}\)
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| The square root of a number: |
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Example |
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Solve:
(i)
\(\begin{aligned} \sqrt{121}&=\sqrt{11\times11} \\\\&=\sqrt{11^2}\\\\&=11. \end{aligned}\)
(ii)
\(\begin{aligned} \sqrt{\dfrac{25}{49}}&=\sqrt{\dfrac{5}{7}\times\dfrac{5}{7}} \\\\&=\sqrt{\bigg(\dfrac{5}{7}\bigg)^2} \\\\&=\dfrac{5}{7}. \end{aligned}\)
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| Generalisation when two square roots are multiplied: |
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Square roots of the same numbers
- The product of two equal square root numbers is the number itself.
- \(\sqrt{a}\times\sqrt{a}=a\)
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Square roots of different numbers
- The product of two different square root numbers is the square root of the product of the two numbers.
- \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\)
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