Question paper, Set 2

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

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

Answer: $y^2$

$23^2 = 23 \times 23$

$= 529$

Answer: $529$

$18^2 + 7^2 = 324 + 49$

$= 373$

Answer: $373$

$40^2 = 40 \times 40 = 1600$

$\therefore$ the number is $40$

Answer: $40$

The number before $5$ is $3$: $3 \times 4 = 12$

Write $25$ after it.

$\therefore 35^2 = 1225$

Answer: $1225$

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

$127^2 = 16129 \Rightarrow \sqrt{16129} = 127$

Answer: $127$

From the table, $\sqrt{729} = 27$ and $\sqrt{256} = 16$

$\sqrt{729} - \sqrt{256} = 27 - 16 = 11$

Answer: $11$

Prime factorization method:

Divide by primes: $3136 \div 2 = 1568$, $1568 \div 2 = 784$, $784 \div 2 = 392$, $392 \div 2 = 196$, $196 \div 2 = 98$, $98 \div 2 = 49$, $49 \div 7 = 7$

$\sqrt{3136} = \sqrt{(2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (7 \times 7)}$

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

$= 56$

Answer: $56$

Long division method:

Pair the digits from the right: $\overline{44}\;\overline{89}$

$6 \times 6 = 36 \le 44$: first digit $6$, and $44 - 36 = 8$

Bring down $89$: we get $889$. New divisor: $6 \times 2 = 12$; $127 \times 7 = 889$ ($128 \times 8 = 1024$ is too big); $889 - 889 = 0$

Remainder $0$. $\therefore \sqrt{4489} = 67$

Answer: $67$

(i) Prime factorization method:

Divide by primes: $2704 \div 2 = 1352$, $1352 \div 2 = 676$, $676 \div 2 = 338$, $338 \div 2 = 169$, $169 \div 13 = 13$

$\sqrt{2704} = \sqrt{(2 \times 2) \times (2 \times 2) \times (13 \times 13)}$

$= 2 \times 2 \times 13$

$= 52$

(ii) Long division method:

Pair the digits from the right: $\overline{27}\;\overline{04}$

$5 \times 5 = 25 \le 27$: first digit $5$, and $27 - 25 = 2$

Bring down $04$: we get $204$. New divisor: $5 \times 2 = 10$; $102 \times 2 = 204$ ($103 \times 3 = 309$ is too big); $204 - 204 = 0$

Remainder $0$. $\therefore \sqrt{2704} = 52$

Answer: $52$

Long division method:

Pair the digits from the right: $\overline{2}\;\overline{68}\;\overline{96}$

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

Bring down $68$: we get $168$. New divisor: $1 \times 2 = 2$; $26 \times 6 = 156$ ($27 \times 7 = 189$ is too big); $168 - 156 = 12$

Bring down $96$: we get $1296$. New divisor: $16 \times 2 = 32$; $324 \times 4 = 1296$ ($325 \times 5 = 1625$ is too big); $1296 - 1296 = 0$

Remainder $0$. $\therefore \sqrt{26896} = 164$

Answer: $164$

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

$\sqrt{5} \times \sqrt{20} = \sqrt{5 \times 20} = \sqrt{100}$

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

Answer: $10$

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

$\therefore \sqrt{13} \times \sqrt{13} = \sqrt{13^2} = 13$

Answer: $13$

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

$\displaystyle \frac{\sqrt{147}}{\sqrt{75}} = \sqrt{\frac{147}{75}} = \sqrt{\frac{3 \times 49}{3 \times 25}}$

$\displaystyle = \sqrt{\frac{49}{25}} = \frac{7}{5}$

Answer: $\displaystyle \frac{7}{5}$

$11 = \sqrt{11} \times \sqrt{11}$

$\displaystyle \frac{11}{\sqrt{11}} = \frac{\sqrt{11} \times \sqrt{11}}{\sqrt{11}} = \sqrt{11}$

Answer: $\sqrt{11}$

LHS $= \sqrt{121} - \sqrt{49} = 11 - 7 = 4 = \sqrt{16}$

RHS $= \sqrt{121 - 49} = \sqrt{72}$

$\therefore$ LHS $\ne$ RHS

Answer: LHS $= 4 = \sqrt{16}$, RHS $= \sqrt{72}$; LHS $\ne$ RHS

(i) LHS $= \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$

RHS $= a + b = 5 + 12 = 17$

$13 \ne 17$, so $\sqrt{a^2 + b^2} \ne a + b$

(ii) LHS $= \sqrt{(5 + 12)^2} = \sqrt{17^2} = 17$

RHS $= 5 + 12 = 17$

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

Answer: (i) $13 \ne 17$ (ii) $17 = 17$

$\displaystyle \sqrt{0.0676} = \sqrt{\frac{676}{10000}}$

$\displaystyle = \frac{\sqrt{676}}{\sqrt{10000}}$

$\displaystyle = \frac{26}{100}$

$= 0.26$

Answer: $0.26$

$\displaystyle \sqrt{0.002025} = \sqrt{\frac{2025}{1000000}}$

$\displaystyle = \frac{\sqrt{2025}}{\sqrt{1000000}}$

$45^2 = 2025$, so $\sqrt{2025} = 45$

$\displaystyle = \frac{45}{1000}$

$= 0.045$

Answer: $0.045$

$\displaystyle \sqrt{46.24} = \sqrt{\frac{4624}{100}} = \frac{\sqrt{4624}}{\sqrt{100}}$

$\sqrt{4624}$ by the long division method:

Pair the digits from the right: $\overline{46}\;\overline{24}$

$6 \times 6 = 36 \le 46$: first digit $6$, and $46 - 36 = 10$

Bring down $24$: we get $1024$. New divisor: $6 \times 2 = 12$; $128 \times 8 = 1024$ ($129 \times 9 = 1161$ is too big); $1024 - 1024 = 0$

$\sqrt{4624} = 68$

$\displaystyle \therefore \sqrt{46.24} = \frac{68}{10} = 6.8$

Answer: $6.8$