Think like a mathematician $\newline$ $6$ a Use a calculator to find $\newline$ i $\quad \sqrt{8} \times \sqrt{2}$ $\newline$ ii $\sqrt{3} \times \sqrt{12}$ $\newline$ iii $\sqrt{20} \times \sqrt{5}$ $\newline$ iv $\sqrt{2} \times \sqrt{18}$ $\newline$ b What do you notice about your answers? $\newline$ c Find another multiplication similar to the multiplications in part a. $\newline$ d Find similar multiplications using cube roots instead of square roots.

Full solution

Q. Think like a mathematician

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6

a Use a calculator to find

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\quad \sqrt{8} \times \sqrt{2}

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\sqrt{3} \times \sqrt{12}

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iii

\sqrt{20} \times \sqrt{5}

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\sqrt{2} \times \sqrt{18}

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b What do you notice about your answers?

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c Find another multiplication similar to the multiplications in part a.

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d Find similar multiplications using cube roots instead of square roots.

Calculate $\sqrt{8} \times \sqrt{2}$ : Calculate $\sqrt{8} \times \sqrt{2}$ . $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$ $\sqrt{8} \times \sqrt{2} = 2\sqrt{2} \times \sqrt{2} = 2 \times 2 = 4$
Calculate $\sqrt{3} \times \sqrt{12}$ : Calculate $\sqrt{3} \times \sqrt{12}$ . $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$ $\sqrt{3} \times \sqrt{12} = \sqrt{3} \times 2\sqrt{3} = 2 \times 3 = 6$
Calculate $\sqrt{20} \times \sqrt{5}$ : Calculate $\sqrt{20} \times \sqrt{5}$ . $\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$ $\sqrt{20} \times \sqrt{5} = 2\sqrt{5} \times \sqrt{5} = 2 \times 5 = 10$
Calculate $\sqrt{2} \times \sqrt{18}$ : Calculate $\sqrt{2} \times \sqrt{18}$ . $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$ $\sqrt{2} \times \sqrt{18} = \sqrt{2} \times 3\sqrt{2} = 3 \times 2 = 6$
Notice the pattern: Notice the pattern. All results are \text{integers}.
Find another multiplication: Find another similar multiplication. $\sqrt{50} \times \sqrt{2} = \sqrt{25 \times 2} \times \sqrt{2} = 5\sqrt{2} \times \sqrt{2} = 5 \times 2 = 10$
Find cube roots multiplications: Find similar multiplications using cube roots. $\sqrt[3]{8} \times \sqrt[3]{2} = 2 \times \sqrt[3]{2}$ $\sqrt[3]{27} \times \sqrt[3]{3} = 3 \times \sqrt[3]{3}$