Question paper, Set 10

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

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

Answer: $q^2$

$13^2 = 13 \times 13$

$= 169$

Answer: $169$

$28^2 - 21^2 = 784 - 441$

$= 343$

Answer: $343$

$100^2 = 100 \times 100 = 10000$

$\therefore$ the number is $100$

Answer: $100$

The number before $5$ is $11$: $11 \times 12 = 132$

Write $25$ after it.

$\therefore 115^2 = 13225$

Answer: $13225$

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

$216^2 = 46656 \Rightarrow \sqrt{46656} = 216$

Answer: $216$

From the table, $\sqrt{576} = 24$ and $\sqrt{361} = 19$

$\sqrt{576} - \sqrt{361} = 24 - 19 = 5$

Answer: $5$

Prime factorization method:

Divide by primes: $2304 \div 2 = 1152$, $1152 \div 2 = 576$, $576 \div 2 = 288$, $288 \div 2 = 144$, $144 \div 2 = 72$, $72 \div 2 = 36$, $36 \div 2 = 18$, $18 \div 2 = 9$, $9 \div 3 = 3$

$\sqrt{2304} = \sqrt{(2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (3 \times 3)}$

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

$= 48$

Answer: $48$

Long division method:

Pair the digits from the right: $\overline{28}\;\overline{09}$

$5 \times 5 = 25 \le 28$: first digit $5$, and $28 - 25 = 3$

Bring down $09$: we get $309$. New divisor: $5 \times 2 = 10$; $103 \times 3 = 309$ ($104 \times 4 = 416$ is too big); $309 - 309 = 0$

Remainder $0$. $\therefore \sqrt{2809} = 53$

Answer: $53$

(i) Prime factorization method:

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

$\sqrt{1089} = \sqrt{(3 \times 3) \times (11 \times 11)}$

$= 3 \times 11$

$= 33$

(ii) Long division method:

Pair the digits from the right: $\overline{10}\;\overline{89}$

$3 \times 3 = 9 \le 10$: first digit $3$, and $10 - 9 = 1$

Bring down $89$: we get $189$. New divisor: $3 \times 2 = 6$; $63 \times 3 = 189$ ($64 \times 4 = 256$ is too big); $189 - 189 = 0$

Remainder $0$. $\therefore \sqrt{1089} = 33$

Answer: $33$

Long division method:

Pair the digits from the right: $\overline{10}\;\overline{49}\;\overline{76}$

$3 \times 3 = 9 \le 10$: first digit $3$, and $10 - 9 = 1$

Bring down $49$: we get $149$. New divisor: $3 \times 2 = 6$; $62 \times 2 = 124$ ($63 \times 3 = 189$ is too big); $149 - 124 = 25$

Bring down $76$: we get $2576$. New divisor: $32 \times 2 = 64$; $644 \times 4 = 2576$ ($645 \times 5 = 3225$ is too big); $2576 - 2576 = 0$

Remainder $0$. $\therefore \sqrt{104976} = 324$

Answer: $324$

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

$\sqrt{5} \times \sqrt{45} = \sqrt{5 \times 45} = \sqrt{225}$

$= \sqrt{15 \times 15} = 15$

Answer: $15$

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

$\therefore \sqrt{14} \times \sqrt{14} = \sqrt{14^2} = 14$

Answer: $14$

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

$\displaystyle \frac{\sqrt{54}}{\sqrt{96}} = \sqrt{\frac{54}{96}} = \sqrt{\frac{6 \times 9}{6 \times 16}}$

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

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

$21 = \sqrt{21} \times \sqrt{21}$

$\displaystyle \frac{21}{\sqrt{21}} = \frac{\sqrt{21} \times \sqrt{21}}{\sqrt{21}} = \sqrt{21}$

Answer: $\sqrt{21}$

LHS $= \sqrt{36} + \sqrt{25} = 6 + 5 = 11 = \sqrt{121}$

RHS $= \sqrt{36 + 25} = \sqrt{61}$

$\therefore$ LHS $\ne$ RHS

Answer: LHS $= 11 = \sqrt{121}$, RHS $= \sqrt{61}$; LHS $\ne$ RHS

(i) LHS $= \sqrt{25^2 - 20^2} = \sqrt{625 - 400} = \sqrt{225} = 15$

RHS $= a - b = 25 - 20 = 5$

$15 \ne 5$, so $\sqrt{a^2 - b^2} \ne a - b$

(ii) LHS $= \sqrt{(25 - 20)^2} = \sqrt{5^2} = 5$

RHS $= 25 - 20 = 5$

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

Answer: (i) $15 \ne 5$ (ii) $5 = 5$

$\displaystyle \sqrt{0.0144} = \sqrt{\frac{144}{10000}}$

$\displaystyle = \frac{\sqrt{144}}{\sqrt{10000}}$

$\displaystyle = \frac{12}{100}$

$= 0.12$

Answer: $0.12$

$\displaystyle \sqrt{0.004096} = \sqrt{\frac{4096}{1000000}}$

$\displaystyle = \frac{\sqrt{4096}}{\sqrt{1000000}}$

$64^2 = 4096$, so $\sqrt{4096} = 64$

$\displaystyle = \frac{64}{1000}$

$= 0.064$

Answer: $0.064$

$\displaystyle \sqrt{24.01} = \sqrt{\frac{2401}{100}} = \frac{\sqrt{2401}}{\sqrt{100}}$

$\sqrt{2401}$ by the long division method:

Pair the digits from the right: $\overline{24}\;\overline{01}$

$4 \times 4 = 16 \le 24$: first digit $4$, and $24 - 16 = 8$

Bring down $01$: we get $801$. New divisor: $4 \times 2 = 8$; $89 \times 9 = 801$; $801 - 801 = 0$

$\sqrt{2401} = 49$

$\displaystyle \therefore \sqrt{24.01} = \frac{49}{10} = 4.9$

Answer: $4.9$