Question paper, Set 9

$74$ is multiplied by itself $2$ times.

$\therefore 74 \times 74 = 74^2$ (read as '74 squared')

Answer: $74^2$

$29^2 = 29 \times 29$

$= 841$

Answer: $841$

$13^2 + 19^2 = 169 + 361$

$= 530$

Answer: $530$

$20^2 = 20 \times 20 = 400$

$\therefore$ the number is $20$

Answer: $20$

The number before $5$ is $10$: $10 \times 11 = 110$

Write $25$ after it.

$\therefore 105^2 = 11025$

Answer: $11025$

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

$191^2 = 36481 \Rightarrow \sqrt{36481} = 191$

Answer: $191$

From the table, $\sqrt{529} = 23$ and $\sqrt{121} = 11$

$\sqrt{529} + \sqrt{121} = 23 + 11 = 34$

Answer: $34$

Prime factorization method:

Divide by primes: $4356 \div 2 = 2178$, $2178 \div 2 = 1089$, $1089 \div 3 = 363$, $363 \div 3 = 121$, $121 \div 11 = 11$

$\sqrt{4356} = \sqrt{(2 \times 2) \times (3 \times 3) \times (11 \times 11)}$

$= 2 \times 3 \times 11$

$= 66$

Answer: $66$

Long division method:

Pair the digits from the right: $\overline{94}\;\overline{09}$

$9 \times 9 = 81 \le 94$: first digit $9$, and $94 - 81 = 13$

Bring down $09$: we get $1309$. New divisor: $9 \times 2 = 18$; $187 \times 7 = 1309$ ($188 \times 8 = 1504$ is too big); $1309 - 1309 = 0$

Remainder $0$. $\therefore \sqrt{9409} = 97$

Answer: $97$

(i) Prime factorization method:

Divide by primes: $6084 \div 2 = 3042$, $3042 \div 2 = 1521$, $1521 \div 3 = 507$, $507 \div 3 = 169$, $169 \div 13 = 13$

$\sqrt{6084} = \sqrt{(2 \times 2) \times (3 \times 3) \times (13 \times 13)}$

$= 2 \times 3 \times 13$

$= 78$

(ii) Long division method:

Pair the digits from the right: $\overline{60}\;\overline{84}$

$7 \times 7 = 49 \le 60$: first digit $7$, and $60 - 49 = 11$

Bring down $84$: we get $1184$. New divisor: $7 \times 2 = 14$; $148 \times 8 = 1184$ ($149 \times 9 = 1341$ is too big); $1184 - 1184 = 0$

Remainder $0$. $\therefore \sqrt{6084} = 78$

Answer: $78$

Long division method:

Pair the digits from the right: $\overline{6}\;\overline{91}\;\overline{69}$

$2 \times 2 = 4 \le 6$: first digit $2$, and $6 - 4 = 2$

Bring down $91$: we get $291$. New divisor: $2 \times 2 = 4$; $46 \times 6 = 276$ ($47 \times 7 = 329$ is too big); $291 - 276 = 15$

Bring down $69$: we get $1569$. New divisor: $26 \times 2 = 52$; $523 \times 3 = 1569$ ($524 \times 4 = 2096$ is too big); $1569 - 1569 = 0$

Remainder $0$. $\therefore \sqrt{69169} = 263$

Answer: $263$

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

$\sqrt{10} \times \sqrt{40} = \sqrt{10 \times 40} = \sqrt{400}$

$= \sqrt{20 \times 20} = 20$

Answer: $20$

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

$\therefore \sqrt{23} \times \sqrt{23} = \sqrt{23^2} = 23$

Answer: $23$

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

$\displaystyle \frac{\sqrt{175}}{\sqrt{112}} = \sqrt{\frac{175}{112}} = \sqrt{\frac{7 \times 25}{7 \times 16}}$

$\displaystyle = \sqrt{\frac{25}{16}} = \frac{5}{4}$

Answer: $\displaystyle \frac{5}{4}$

$19 = \sqrt{19} \times \sqrt{19}$

$\displaystyle \frac{19}{\sqrt{19}} = \frac{\sqrt{19} \times \sqrt{19}}{\sqrt{19}} = \sqrt{19}$

Answer: $\sqrt{19}$

LHS $= \sqrt{144} - \sqrt{25} = 12 - 5 = 7 = \sqrt{49}$

RHS $= \sqrt{144 - 25} = \sqrt{119}$

$\therefore$ LHS $\ne$ RHS

Answer: LHS $= 7 = \sqrt{49}$, RHS $= \sqrt{119}$; LHS $\ne$ RHS

(i) LHS $= \sqrt{26^2 - 24^2} = \sqrt{676 - 576} = \sqrt{100} = 10$

RHS $= a - b = 26 - 24 = 2$

$10 \ne 2$, so $\sqrt{a^2 - b^2} \ne a - b$

(ii) LHS $= \sqrt{(26 - 24)^2} = \sqrt{2^2} = 2$

RHS $= 26 - 24 = 2$

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

Answer: (i) $10 \ne 2$ (ii) $2 = 2$

$\displaystyle \sqrt{0.0256} = \sqrt{\frac{256}{10000}}$

$\displaystyle = \frac{\sqrt{256}}{\sqrt{10000}}$

$\displaystyle = \frac{16}{100}$

$= 0.16$

Answer: $0.16$

$\displaystyle \sqrt{0.001936} = \sqrt{\frac{1936}{1000000}}$

$\displaystyle = \frac{\sqrt{1936}}{\sqrt{1000000}}$

$44^2 = 1936$, so $\sqrt{1936} = 44$

$\displaystyle = \frac{44}{1000}$

$= 0.044$

Answer: $0.044$

$\displaystyle \sqrt{26.01} = \sqrt{\frac{2601}{100}} = \frac{\sqrt{2601}}{\sqrt{100}}$

$\sqrt{2601}$ by the long division method:

Pair the digits from the right: $\overline{26}\;\overline{01}$

$5 \times 5 = 25 \le 26$: first digit $5$, and $26 - 25 = 1$

Bring down $01$: we get $101$. New divisor: $5 \times 2 = 10$; $101 \times 1 = 101$ ($102 \times 2 = 204$ is too big); $101 - 101 = 0$

$\sqrt{2601} = 51$

$\displaystyle \therefore \sqrt{26.01} = \frac{51}{10} = 5.1$

Answer: $5.1$