[5 marks sum]

Questions

Question 15 Marks

Evaluate $\sqrt{1 \frac{4}{5} \times 14 \frac{21}{22} \times 2 \frac{7}{55}}$

Answer

$\sqrt{1 \frac{4}{5} \times 14 \frac{21}{22} \times 2 \frac{7}{55}}$
$=\sqrt{\frac{9}{5} \times \frac{637}{44} \times \frac{117}{55}}$
$=\sqrt{\frac{9 \times 637 \times 117}{5 \times 44 \times 5}}$
$=\sqrt{\frac{9 \times 7 \times 7 \times 13 \times 13 \times 9}{5 \times 11 \times 2 \times 2 \times 11 \times 55}}$

$7$	$637$
$7$	$91$
	$13$

$9$	$117$
	$13$

$=\frac{9 \times 7 \times 13}{5 \times 11 \times 2}$
$=\frac{819}{110}$
$=7 \frac{49}{110}$

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Question 25 Marks

Evaluate $\sqrt{\frac{25}{32} \times 2 \frac{13}{18} \times 0.25}$

Answer

$\sqrt{\frac{25}{32} \times 2 \frac{13}{18} \times 0.25}$
$=\sqrt{\frac{25}{32} \times \frac{49}{18} \times 0.25}$
$=\sqrt{\frac{25}{32} \times \frac{49}{18} \times \frac{25}{100}}$
$=\sqrt{\frac{25 \times 49 \times 25}{32 \times 18 \times 100}}$
$=\sqrt{\frac{25 \times 49 \times 1}{32 \times 18 \times 4}}$
$=\sqrt{\frac{5 \times 5 \times 7 \times 7}{(2 \times 2 \times 2 \times 2 \times 2) \times(2 \times 3 \times 3) \times(2 \times 2)}}$
$=\sqrt{\frac{\overline{5 \times 5} \times \overline{7 \times 7}}{\overline{2 \times 2} \times \overline{2 \times 2} \times \overline{2 \times 2} \times \overline{3 \times 3} \times \overline{2 \times 2}}}$
$=\frac{5 \times 7}{2 \times 2 \times 2 \times 3 \times 2}$
$=\frac{35}{48}$

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Question 35 Marks

Find the smallest number by which $2592$ be multiplied so that the product is a perfect square.

Answer

$2592=\overline{2 \times 2} \times \overline{2 \times 2} \times 2 \times \overline{3 \times 3} \times \overline{3 \times 3}$
On grouping the prime factors of $2592$ as shown; on factor i.e. $2$ is left which cannot be paired with equal factor.

$2$	$2592$
$2$	$1296$
$2$	$648$
$2$	$324$
$2$	$162$
$3$	$81$
$3$	$27$
$3$	$9$
$3$	$3$
	$1$

$2$ is the smallest number that must be multiplied by $2592$ to get a perfect square.
$\sqrt{2592 \times 2}=\sqrt{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3}$
$\sqrt{5184}$
$=2 \times 2 \times 2 \times 3 \times 3$
$=8 \times 9$
$=72$

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