Question paper, Set 3

$43$ is multiplied by itself $2$ times.

$\therefore 43 \times 43 = 43^2$ (read as '43 squared')

Answer: $43^2$

$14^2 = 14 \times 14$

$= 196$

Answer: $196$

$23^2 - 18^2 = 529 - 324$

$= 205$

Answer: $205$

$90^2 = 90 \times 90 = 8100$

$\therefore$ the number is $90$

Answer: $90$

The number before $5$ is $8$: $8 \times 9 = 72$

Write $25$ after it.

$\therefore 85^2 = 7225$

Answer: $7225$

$a^2 = b \Rightarrow \sqrt{b} = a$

$132^2 = 17424 \Rightarrow \sqrt{17424} = 132$

Answer: $132$

From the table, $\sqrt{169} = 13$ and $\sqrt{576} = 24$

$\sqrt{169} + \sqrt{576} = 13 + 24 = 37$

Answer: $37$

Prime factorization method:

Divide by primes: $5184 \div 2 = 2592$, $2592 \div 2 = 1296$, $1296 \div 2 = 648$, $648 \div 2 = 324$, $324 \div 2 = 162$, $162 \div 2 = 81$, $81 \div 3 = 27$, $27 \div 3 = 9$, $9 \div 3 = 3$

$\sqrt{5184} = \sqrt{(2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (3 \times 3) \times (3 \times 3)}$

$= 2 \times 2 \times 2 \times 3 \times 3$

$= 72$

Answer: $72$

Long division method:

Pair the digits from the right: $\overline{79}\;\overline{21}$

$8 \times 8 = 64 \le 79$: first digit $8$, and $79 - 64 = 15$

Bring down $21$: we get $1521$. New divisor: $8 \times 2 = 16$; $169 \times 9 = 1521$; $1521 - 1521 = 0$

Remainder $0$. $\therefore \sqrt{7921} = 89$

Answer: $89$

(i) Prime factorization method:

Divide by primes: $5929 \div 7 = 847$, $847 \div 7 = 121$, $121 \div 11 = 11$

$\sqrt{5929} = \sqrt{(7 \times 7) \times (11 \times 11)}$

$= 7 \times 11$

$= 77$

(ii) Long division method:

Pair the digits from the right: $\overline{59}\;\overline{29}$

$7 \times 7 = 49 \le 59$: first digit $7$, and $59 - 49 = 10$

Bring down $29$: we get $1029$. New divisor: $7 \times 2 = 14$; $147 \times 7 = 1029$ ($148 \times 8 = 1184$ is too big); $1029 - 1029 = 0$

Remainder $0$. $\therefore \sqrt{5929} = 77$

Answer: $77$

Long division method:

Pair the digits from the right: $\overline{8}\;\overline{35}\;\overline{21}$

$2 \times 2 = 4 \le 8$: first digit $2$, and $8 - 4 = 4$

Bring down $35$: we get $435$. New divisor: $2 \times 2 = 4$; $48 \times 8 = 384$ ($49 \times 9 = 441$ is too big); $435 - 384 = 51$

Bring down $21$: we get $5121$. New divisor: $28 \times 2 = 56$; $569 \times 9 = 5121$; $5121 - 5121 = 0$

Remainder $0$. $\therefore \sqrt{83521} = 289$

Answer: $289$

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

$\sqrt{6} \times \sqrt{24} = \sqrt{6 \times 24} = \sqrt{144}$

$= \sqrt{12 \times 12} = 12$

Answer: $12$

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

$\therefore \sqrt{5} \times \sqrt{5} = \sqrt{5^2} = 5$

Answer: $5$

$\displaystyle \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

$\displaystyle \frac{\sqrt{48}}{\sqrt{27}} = \sqrt{\frac{48}{27}} = \sqrt{\frac{3 \times 16}{3 \times 9}}$

$\displaystyle = \sqrt{\frac{16}{9}} = \frac{4}{3}$

Answer: $\displaystyle \frac{4}{3}$

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

$\displaystyle \frac{6}{\sqrt{6}} = \frac{\sqrt{6} \times \sqrt{6}}{\sqrt{6}} = \sqrt{6}$

Answer: $\sqrt{6}$

LHS $= \sqrt{225} + \sqrt{400} = 15 + 20 = 35$

RHS $= \sqrt{225 + 400} = \sqrt{625} = 25$

$\therefore$ LHS $\ne$ RHS

Answer: LHS $= 35$, RHS $= 25$; LHS $\ne$ RHS

(i) LHS $= \sqrt{35^2 - 28^2} = \sqrt{1225 - 784} = \sqrt{441} = 21$

RHS $= a - b = 35 - 28 = 7$

$21 \ne 7$, so $\sqrt{a^2 - b^2} \ne a - b$

(ii) LHS $= \sqrt{(35 - 28)^2} = \sqrt{7^2} = 7$

RHS $= 35 - 28 = 7$

LHS $=$ RHS, so $\sqrt{(a - b)^2} = a - b$

Answer: (i) $21 \ne 7$ (ii) $7 = 7$

$\displaystyle \sqrt{0.0289} = \sqrt{\frac{289}{10000}}$

$\displaystyle = \frac{\sqrt{289}}{\sqrt{10000}}$

$\displaystyle = \frac{17}{100}$

$= 0.17$

Answer: $0.17$

$\displaystyle \sqrt{0.003025} = \sqrt{\frac{3025}{1000000}}$

$\displaystyle = \frac{\sqrt{3025}}{\sqrt{1000000}}$

$55^2 = 3025$, so $\sqrt{3025} = 55$

$\displaystyle = \frac{55}{1000}$

$= 0.055$

Answer: $0.055$

$\displaystyle \sqrt{84.64} = \sqrt{\frac{8464}{100}} = \frac{\sqrt{8464}}{\sqrt{100}}$

$\sqrt{8464}$ by the long division method:

Pair the digits from the right: $\overline{84}\;\overline{64}$

$9 \times 9 = 81 \le 84$: first digit $9$, and $84 - 81 = 3$

Bring down $64$: we get $364$. New divisor: $9 \times 2 = 18$; $182 \times 2 = 364$ ($183 \times 3 = 549$ is too big); $364 - 364 = 0$

$\sqrt{8464} = 92$

$\displaystyle \therefore \sqrt{84.64} = \frac{92}{10} = 9.2$

Answer: $9.2$