Question paper, Set 4

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

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

Answer: $m^2$

$28^2 = 28 \times 28$

$= 784$

Answer: $784$

$26^2 - 11^2 = 676 - 121$

$= 555$

Answer: $555$

$60^2 = 60 \times 60 = 3600$

$\therefore$ the number is $60$

Answer: $60$

The number before $5$ is $4$: $4 \times 5 = 20$

Write $25$ after it.

$\therefore 45^2 = 2025$

Answer: $2025$

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

$146^2 = 21316 \Rightarrow \sqrt{21316} = 146$

Answer: $146$

From the table, $\sqrt{441} = 21$ and $\sqrt{289} = 17$

$\sqrt{441} + \sqrt{289} = 21 + 17 = 38$

Answer: $38$

Prime factorization method:

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

$\sqrt{1296} = \sqrt{(2 \times 2) \times (2 \times 2) \times (3 \times 3) \times (3 \times 3)}$

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

$= 36$

Answer: $36$

Long division method:

Pair the digits from the right: $\overline{88}\;\overline{36}$

$9 \times 9 = 81 \le 88$: first digit $9$, and $88 - 81 = 7$

Bring down $36$: we get $736$. New divisor: $9 \times 2 = 18$; $184 \times 4 = 736$ ($185 \times 5 = 925$ is too big); $736 - 736 = 0$

Remainder $0$. $\therefore \sqrt{8836} = 94$

Answer: $94$

(i) Prime factorization method:

Divide by primes: $7744 \div 2 = 3872$, $3872 \div 2 = 1936$, $1936 \div 2 = 968$, $968 \div 2 = 484$, $484 \div 2 = 242$, $242 \div 2 = 121$, $121 \div 11 = 11$

$\sqrt{7744} = \sqrt{(2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (11 \times 11)}$

$= 2 \times 2 \times 2 \times 11$

$= 88$

(ii) Long division method:

Pair the digits from the right: $\overline{77}\;\overline{44}$

$8 \times 8 = 64 \le 77$: first digit $8$, and $77 - 64 = 13$

Bring down $44$: we get $1344$. New divisor: $8 \times 2 = 16$; $168 \times 8 = 1344$ ($169 \times 9 = 1521$ is too big); $1344 - 1344 = 0$

Remainder $0$. $\therefore \sqrt{7744} = 88$

Answer: $88$

Long division method:

Pair the digits from the right: $\overline{17}\;\overline{05}\;\overline{69}$

$4 \times 4 = 16 \le 17$: first digit $4$, and $17 - 16 = 1$

Bring down $05$: we get $105$. New divisor: $4 \times 2 = 8$; $81 \times 1 = 81$ ($82 \times 2 = 164$ is too big); $105 - 81 = 24$

Bring down $69$: we get $2469$. New divisor: $41 \times 2 = 82$; $823 \times 3 = 2469$ ($824 \times 4 = 3296$ is too big); $2469 - 2469 = 0$

Remainder $0$. $\therefore \sqrt{170569} = 413$

Answer: $413$

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

$\sqrt{2} \times \sqrt{32} = \sqrt{2 \times 32} = \sqrt{64}$

$= \sqrt{8 \times 8} = 8$

Answer: $8$

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

$\therefore \sqrt{19} \times \sqrt{19} = \sqrt{19^2} = 19$

Answer: $19$

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

$\displaystyle \frac{\sqrt{125}}{\sqrt{45}} = \sqrt{\frac{125}{45}} = \sqrt{\frac{5 \times 25}{5 \times 9}}$

$\displaystyle = \sqrt{\frac{25}{9}} = \frac{5}{3}$

Answer: $\displaystyle \frac{5}{3}$

$13 = \sqrt{13} \times \sqrt{13}$

$\displaystyle \frac{13}{\sqrt{13}} = \frac{\sqrt{13} \times \sqrt{13}}{\sqrt{13}} = \sqrt{13}$

Answer: $\sqrt{13}$

LHS $= \sqrt{196} - \sqrt{81} = 14 - 9 = 5 = \sqrt{25}$

RHS $= \sqrt{196 - 81} = \sqrt{115}$

$\therefore$ LHS $\ne$ RHS

Answer: LHS $= 5 = \sqrt{25}$, RHS $= \sqrt{115}$; LHS $\ne$ RHS

(i) LHS $= \sqrt{15^2 - 9^2} = \sqrt{225 - 81} = \sqrt{144} = 12$

RHS $= a - b = 15 - 9 = 6$

$12 \ne 6$, so $\sqrt{a^2 - b^2} \ne a - b$

(ii) LHS $= \sqrt{(15 - 9)^2} = \sqrt{6^2} = 6$

RHS $= 15 - 9 = 6$

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

Answer: (i) $12 \ne 6$ (ii) $6 = 6$

$\displaystyle \sqrt{0.0169} = \sqrt{\frac{169}{10000}}$

$\displaystyle = \frac{\sqrt{169}}{\sqrt{10000}}$

$\displaystyle = \frac{13}{100}$

$= 0.13$

Answer: $0.13$

$\displaystyle \sqrt{0.004225} = \sqrt{\frac{4225}{1000000}}$

$\displaystyle = \frac{\sqrt{4225}}{\sqrt{1000000}}$

$65^2 = 4225$, so $\sqrt{4225} = 65$

$\displaystyle = \frac{65}{1000}$

$= 0.065$

Answer: $0.065$

$\displaystyle \sqrt{50.41} = \sqrt{\frac{5041}{100}} = \frac{\sqrt{5041}}{\sqrt{100}}$

$\sqrt{5041}$ by the long division method:

Pair the digits from the right: $\overline{50}\;\overline{41}$

$7 \times 7 = 49 \le 50$: first digit $7$, and $50 - 49 = 1$

Bring down $41$: we get $141$. New divisor: $7 \times 2 = 14$; $141 \times 1 = 141$ ($142 \times 2 = 284$ is too big); $141 - 141 = 0$

$\sqrt{5041} = 71$

$\displaystyle \therefore \sqrt{50.41} = \frac{71}{10} = 7.1$

Answer: $7.1$