Question paper, Set 5

$52$ is multiplied by itself $2$ times.

$\therefore 52 \times 52 = 52^2$ (read as '52 squared')

Answer: $52^2$

$19^2 = 19 \times 19$

$= 361$

Answer: $361$

$24^2 + 9^2 = 576 + 81$

$= 657$

Answer: $657$

$80^2 = 80 \times 80 = 6400$

$\therefore$ the number is $80$

Answer: $80$

The number before $5$ is $9$: $9 \times 10 = 90$

Write $25$ after it.

$\therefore 95^2 = 9025$

Answer: $9025$

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

$158^2 = 24964 \Rightarrow \sqrt{24964} = 158$

Answer: $158$

From the table, $\sqrt{841} = 29$ and $\sqrt{121} = 11$

$\sqrt{841} - \sqrt{121} = 29 - 11 = 18$

Answer: $18$

Prime factorization method:

Divide by primes: $7056 \div 2 = 3528$, $3528 \div 2 = 1764$, $1764 \div 2 = 882$, $882 \div 2 = 441$, $441 \div 3 = 147$, $147 \div 3 = 49$, $49 \div 7 = 7$

$\sqrt{7056} = \sqrt{(2 \times 2) \times (2 \times 2) \times (3 \times 3) \times (7 \times 7)}$

$= 2 \times 2 \times 3 \times 7$

$= 84$

Answer: $84$

Long division method:

Pair the digits from the right: $\overline{22}\;\overline{09}$

$4 \times 4 = 16 \le 22$: first digit $4$, and $22 - 16 = 6$

Bring down $09$: we get $609$. New divisor: $4 \times 2 = 8$; $87 \times 7 = 609$ ($88 \times 8 = 704$ is too big); $609 - 609 = 0$

Remainder $0$. $\therefore \sqrt{2209} = 47$

Answer: $47$

(i) Prime factorization method:

Divide by primes: $8281 \div 7 = 1183$, $1183 \div 7 = 169$, $169 \div 13 = 13$

$\sqrt{8281} = \sqrt{(7 \times 7) \times (13 \times 13)}$

$= 7 \times 13$

$= 91$

(ii) Long division method:

Pair the digits from the right: $\overline{82}\;\overline{81}$

$9 \times 9 = 81 \le 82$: first digit $9$, and $82 - 81 = 1$

Bring down $81$: we get $181$. New divisor: $9 \times 2 = 18$; $181 \times 1 = 181$ ($182 \times 2 = 364$ is too big); $181 - 181 = 0$

Remainder $0$. $\therefore \sqrt{8281} = 91$

Answer: $91$

Long division method:

Pair the digits from the right: $\overline{12}\;\overline{67}\;\overline{36}$

$3 \times 3 = 9 \le 12$: first digit $3$, and $12 - 9 = 3$

Bring down $67$: we get $367$. New divisor: $3 \times 2 = 6$; $65 \times 5 = 325$ ($66 \times 6 = 396$ is too big); $367 - 325 = 42$

Bring down $36$: we get $4236$. New divisor: $35 \times 2 = 70$; $706 \times 6 = 4236$ ($707 \times 7 = 4949$ is too big); $4236 - 4236 = 0$

Remainder $0$. $\therefore \sqrt{126736} = 356$

Answer: $356$

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

$\sqrt{7} \times \sqrt{28} = \sqrt{7 \times 28} = \sqrt{196}$

$= \sqrt{14 \times 14} = 14$

Answer: $14$

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

$\therefore \sqrt{11} \times \sqrt{11} = \sqrt{11^2} = 11$

Answer: $11$

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

$\displaystyle \frac{\sqrt{98}}{\sqrt{162}} = \sqrt{\frac{98}{162}} = \sqrt{\frac{2 \times 49}{2 \times 81}}$

$\displaystyle = \sqrt{\frac{49}{81}} = \frac{7}{9}$

Answer: $\displaystyle \frac{7}{9}$

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

$\displaystyle \frac{10}{\sqrt{10}} = \frac{\sqrt{10} \times \sqrt{10}}{\sqrt{10}} = \sqrt{10}$

Answer: $\sqrt{10}$

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

RHS $= \sqrt{81 + 25} = \sqrt{106}$

$\therefore$ LHS $\ne$ RHS

Answer: LHS $= 14 = \sqrt{196}$, RHS $= \sqrt{106}$; LHS $\ne$ RHS

(i) LHS $= \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17$

RHS $= a + b = 8 + 15 = 23$

$17 \ne 23$, so $\sqrt{a^2 + b^2} \ne a + b$

(ii) LHS $= \sqrt{(8 + 15)^2} = \sqrt{23^2} = 23$

RHS $= 8 + 15 = 23$

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

Answer: (i) $17 \ne 23$ (ii) $23 = 23$

$\displaystyle \sqrt{0.0784} = \sqrt{\frac{784}{10000}}$

$\displaystyle = \frac{\sqrt{784}}{\sqrt{10000}}$

$\displaystyle = \frac{28}{100}$

$= 0.28$

Answer: $0.28$

$\displaystyle \sqrt{0.005625} = \sqrt{\frac{5625}{1000000}}$

$\displaystyle = \frac{\sqrt{5625}}{\sqrt{1000000}}$

$75^2 = 5625$, so $\sqrt{5625} = 75$

$\displaystyle = \frac{75}{1000}$

$= 0.075$

Answer: $0.075$

$\displaystyle \sqrt{14.44} = \sqrt{\frac{1444}{100}} = \frac{\sqrt{1444}}{\sqrt{100}}$

$\sqrt{1444}$ by the long division method:

Pair the digits from the right: $\overline{14}\;\overline{44}$

$3 \times 3 = 9 \le 14$: first digit $3$, and $14 - 9 = 5$

Bring down $44$: we get $544$. New divisor: $3 \times 2 = 6$; $68 \times 8 = 544$ ($69 \times 9 = 621$ is too big); $544 - 544 = 0$

$\sqrt{1444} = 38$

$\displaystyle \therefore \sqrt{14.44} = \frac{38}{10} = 3.8$

Answer: $3.8$