Surds — Multiplying & Dividing (Ex 4C)

Year 10 Mathematics Core · Cambridge Ch 4, §4C · Mr Wong

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Class

Date

Learning intentions.

Multiply two surds using $\sqrt{x}\times\sqrt{y}=\sqrt{xy}$ and $a\sqrt{x}\times b\sqrt{y}=ab\sqrt{xy}$
Divide surds using $\dfrac{\sqrt{x}}{\sqrt{y}}=\sqrt{\dfrac{x}{y}}$ and use $\sqrt{x}\times\sqrt{x}=x$
Apply the distributive law to brackets containing surds, simplifying the result

Key results

$\sqrt{x}\;\times\;\sqrt{y}\;=\;\sqrt{xy}\qquad a\sqrt{x}\;\times\;b\sqrt{y}\;=\;ab\sqrt{xy}\qquad \dfrac{a\sqrt{x}}{b\sqrt{y}}\;=\;\dfrac{a}{b}\sqrt{\dfrac{x}{y}}\qquad \sqrt{x}\;\times\;\sqrt{x}\;=\;x$

Warm-up — fill in the gaps

1. State the missing parts.

a $\sqrt{15}\div\sqrt{3}=\sqrt{\underline{\phantom{xx}}}=\sqrt{\underline{\phantom{x}}}$

b $\sqrt{42}\div\sqrt{7}=\sqrt{\underline{\phantom{xx}}}=\sqrt{\underline{\phantom{x}}}$

c $\sqrt{6}\times\sqrt{5}=\sqrt{6\times\underline{\phantom{x}}}=\sqrt{\underline{\phantom{xx}}}$

d $\sqrt{11}\times\sqrt{2}=\sqrt{11\times\underline{\phantom{x}}}=\sqrt{\underline{\phantom{xx}}}$

2. Use $\sqrt{x}\times\sqrt{x}=x$ to evaluate.

a $\sqrt{6}\times\sqrt{6}=$

b $\sqrt{7}^{\,2}=$

c $(\sqrt{5})^{2}=$

d $(\sqrt{13})^{2}=$

Part A — Multiplying surds

3. Multiply and simplify.

a $\sqrt{3}\times\sqrt{7}=$

b $\sqrt{10}\times\sqrt{2}=$

c $4\sqrt{2}\times\sqrt{5}=$

d $3\sqrt{2}\times 5\sqrt{3}=$

e $2\sqrt{6}\times 5\sqrt{2}=$

f $3\sqrt{5}\times 2\sqrt{10}=$

4. Squares of surds — give whole-number answers.

a $(\sqrt{6})^{2}=$

b $(\sqrt{11})^{2}=$

c $(3\sqrt{2})^{2}=$

d $(4\sqrt{5})^{2}=$

Surds 4C — continued

Part B (dividing) · Part C (distributive law) · Challenge

Name

Part B — Dividing surds

5. Divide and simplify.

a $\sqrt{20}\div\sqrt{5}=$

b $\sqrt{30}\div\sqrt{6}=$

c $\sqrt{72}\div\sqrt{8}=$

d $\dfrac{10\sqrt{24}}{5\sqrt{6}}=$

e $\dfrac{8\sqrt{45}}{2\sqrt{5}}=$

f $\dfrac{6\sqrt{50}}{3\sqrt{2}}=$

Part C — Distributive law

6. Expand and simplify each. Show the multiplication-out step before simplifying surds.

a $\sqrt{2}\big(\sqrt{5}+\sqrt{7}\big)$

b $\sqrt{3}\big(2\sqrt{3}-\sqrt{5}\big)$

c $2\sqrt{5}\big(\sqrt{10}+3\sqrt{5}\big)$

d $\sqrt{6}\big(\sqrt{2}+\sqrt{3}\big)$

Challenge

7. A rectangle has length $3\sqrt{6}$ cm and width $2\sqrt{2}$ cm. (a) Find its area as a simplified surd. (b) Find its perimeter in surd form. (c) Find the length of the diagonal using Pythagoras' theorem, and simplify your answer.

8. Show that $\big(3\sqrt{2}+\sqrt{5}\big)\big(3\sqrt{2}-\sqrt{5}\big)$ is a whole number, and state its value.