Question paper, Set 1

$37$ is multiplied by itself $2$ times.

$\therefore 37 \times 37 = 37^2$ (read as '37 squared')

Answer: $37^2$

$17^2 = 17 \times 17$

$= 289$

Answer: $289$

$11^2 + 16^2 = 121 + 256$

$= 377$

Answer: $377$

$70^2 = 70 \times 70 = 4900$

$\therefore$ the number is $70$

Answer: $70$

The number before $5$ is $6$: $6 \times 7 = 42$

Write $25$ after it.

$\therefore 65^2 = 4225$

Answer: $4225$

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

$113^2 = 12769 \Rightarrow \sqrt{12769} = 113$

Answer: $113$

From the table, $\sqrt{361} = 19$ and $\sqrt{144} = 12$

$\sqrt{361} + \sqrt{144} = 19 + 12 = 31$

Answer: $31$

Prime factorization method:

Divide by primes: $1764 \div 2 = 882$, $882 \div 2 = 441$, $441 \div 3 = 147$, $147 \div 3 = 49$, $49 \div 7 = 7$

$\sqrt{1764} = \sqrt{(2 \times 2) \times (3 \times 3) \times (7 \times 7)}$

$= 2 \times 3 \times 7$

$= 42$

Answer: $42$

Long division method:

Pair the digits from the right: $\overline{53}\;\overline{29}$

$7 \times 7 = 49 \le 53$: first digit $7$, and $53 - 49 = 4$

Bring down $29$: we get $429$. New divisor: $7 \times 2 = 14$; $143 \times 3 = 429$ ($144 \times 4 = 576$ is too big); $429 - 429 = 0$

Remainder $0$. $\therefore \sqrt{5329} = 73$

Answer: $73$

(i) Prime factorization method:

Divide by primes: $1936 \div 2 = 968$, $968 \div 2 = 484$, $484 \div 2 = 242$, $242 \div 2 = 121$, $121 \div 11 = 11$

$\sqrt{1936} = \sqrt{(2 \times 2) \times (2 \times 2) \times (11 \times 11)}$

$= 2 \times 2 \times 11$

$= 44$

(ii) Long division method:

Pair the digits from the right: $\overline{19}\;\overline{36}$

$4 \times 4 = 16 \le 19$: first digit $4$, and $19 - 16 = 3$

Bring down $36$: we get $336$. New divisor: $4 \times 2 = 8$; $84 \times 4 = 336$ ($85 \times 5 = 425$ is too big); $336 - 336 = 0$

Remainder $0$. $\therefore \sqrt{1936} = 44$

Answer: $44$

Long division method:

Pair the digits from the right: $\overline{5}\;\overline{61}\;\overline{69}$

$2 \times 2 = 4 \le 5$: first digit $2$, and $5 - 4 = 1$

Bring down $61$: we get $161$. New divisor: $2 \times 2 = 4$; $43 \times 3 = 129$ ($44 \times 4 = 176$ is too big); $161 - 129 = 32$

Bring down $69$: we get $3269$. New divisor: $23 \times 2 = 46$; $467 \times 7 = 3269$ ($468 \times 8 = 3744$ is too big); $3269 - 3269 = 0$

Remainder $0$. $\therefore \sqrt{56169} = 237$

Answer: $237$

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

$\sqrt{3} \times \sqrt{27} = \sqrt{3 \times 27} = \sqrt{81}$

$= \sqrt{9 \times 9} = 9$

Answer: $9$

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

$\therefore \sqrt{7} \times \sqrt{7} = \sqrt{7^2} = 7$

Answer: $7$

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

$\displaystyle \frac{\sqrt{50}}{\sqrt{72}} = \sqrt{\frac{50}{72}} = \sqrt{\frac{2 \times 25}{2 \times 36}}$

$\displaystyle = \sqrt{\frac{25}{36}} = \frac{5}{6}$

Answer: $\displaystyle \frac{5}{6}$

$5 = \sqrt{5} \times \sqrt{5}$

$\displaystyle \frac{5}{\sqrt{5}} = \frac{\sqrt{5} \times \sqrt{5}}{\sqrt{5}} = \sqrt{5}$

Answer: $\sqrt{5}$

LHS $= \sqrt{324} + \sqrt{576} = 18 + 24 = 42$

RHS $= \sqrt{324 + 576} = \sqrt{900} = 30$

$\therefore$ LHS $\ne$ RHS

Answer: LHS $= 42$, RHS $= 30$; LHS $\ne$ RHS

(i) LHS $= \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8$

RHS $= a - b = 10 - 6 = 4$

$8 \ne 4$, so $\sqrt{a^2 - b^2} \ne a - b$

(ii) LHS $= \sqrt{(10 - 6)^2} = \sqrt{4^2} = 4$

RHS $= 10 - 6 = 4$

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

Answer: (i) $8 \ne 4$ (ii) $4 = 4$

$\displaystyle \sqrt{0.0441} = \sqrt{\frac{441}{10000}}$

$\displaystyle = \frac{\sqrt{441}}{\sqrt{10000}}$

$\displaystyle = \frac{21}{100}$

$= 0.21$

Answer: $0.21$

$\displaystyle \sqrt{0.001225} = \sqrt{\frac{1225}{1000000}}$

$\displaystyle = \frac{\sqrt{1225}}{\sqrt{1000000}}$

$35^2 = 1225$, so $\sqrt{1225} = 35$

$\displaystyle = \frac{35}{1000}$

$= 0.035$

Answer: $0.035$

$\displaystyle \sqrt{54.76} = \sqrt{\frac{5476}{100}} = \frac{\sqrt{5476}}{\sqrt{100}}$

$\sqrt{5476}$ by the long division method:

Pair the digits from the right: $\overline{54}\;\overline{76}$

$7 \times 7 = 49 \le 54$: first digit $7$, and $54 - 49 = 5$

Bring down $76$: we get $576$. New divisor: $7 \times 2 = 14$; $144 \times 4 = 576$ ($145 \times 5 = 725$ is too big); $576 - 576 = 0$

$\sqrt{5476} = 74$

$\displaystyle \therefore \sqrt{54.76} = \frac{74}{10} = 7.4$

Answer: $7.4$