Question paper, Set 7

$61$ is multiplied by itself $2$ times.

$\therefore 61 \times 61 = 61^2$ (read as '61 squared')

Answer: $61^2$

$12^2 = 12 \times 12$

$= 144$

Answer: $144$

$21^2 + 8^2 = 441 + 64$

$= 505$

Answer: $505$

$30^2 = 30 \times 30 = 900$

$\therefore$ the number is $30$

Answer: $30$

The number before $5$ is $5$: $5 \times 6 = 30$

Write $25$ after it.

$\therefore 55^2 = 3025$

Answer: $3025$

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

$174^2 = 30276 \Rightarrow \sqrt{30276} = 174$

Answer: $174$

From the table, $\sqrt{676} = 26$ and $\sqrt{196} = 14$

$\sqrt{676} - \sqrt{196} = 26 - 14 = 12$

Answer: $12$

Prime factorization method:

Divide by primes: $9216 \div 2 = 4608$, $4608 \div 2 = 2304$, $2304 \div 2 = 1152$, $1152 \div 2 = 576$, $576 \div 2 = 288$, $288 \div 2 = 144$, $144 \div 2 = 72$, $72 \div 2 = 36$, $36 \div 2 = 18$, $18 \div 2 = 9$, $9 \div 3 = 3$

$\sqrt{9216} = \sqrt{(2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (3 \times 3)}$

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

$= 96$

Answer: $96$

Long division method:

Pair the digits from the right: $\overline{73}\;\overline{96}$

$8 \times 8 = 64 \le 73$: first digit $8$, and $73 - 64 = 9$

Bring down $96$: we get $996$. New divisor: $8 \times 2 = 16$; $166 \times 6 = 996$ ($167 \times 7 = 1169$ is too big); $996 - 996 = 0$

Remainder $0$. $\therefore \sqrt{7396} = 86$

Answer: $86$

(i) Prime factorization method:

Divide by primes: $9801 \div 3 = 3267$, $3267 \div 3 = 1089$, $1089 \div 3 = 363$, $363 \div 3 = 121$, $121 \div 11 = 11$

$\sqrt{9801} = \sqrt{(3 \times 3) \times (3 \times 3) \times (11 \times 11)}$

$= 3 \times 3 \times 11$

$= 99$

(ii) Long division method:

Pair the digits from the right: $\overline{98}\;\overline{01}$

$9 \times 9 = 81 \le 98$: first digit $9$, and $98 - 81 = 17$

Bring down $01$: we get $1701$. New divisor: $9 \times 2 = 18$; $189 \times 9 = 1701$; $1701 - 1701 = 0$

Remainder $0$. $\therefore \sqrt{9801} = 99$

Answer: $99$

Long division method:

Pair the digits from the right: $\overline{7}\;\overline{72}\;\overline{84}$

$2 \times 2 = 4 \le 7$: first digit $2$, and $7 - 4 = 3$

Bring down $72$: we get $372$. New divisor: $2 \times 2 = 4$; $47 \times 7 = 329$ ($48 \times 8 = 384$ is too big); $372 - 329 = 43$

Bring down $84$: we get $4384$. New divisor: $27 \times 2 = 54$; $548 \times 8 = 4384$ ($549 \times 9 = 4941$ is too big); $4384 - 4384 = 0$

Remainder $0$. $\therefore \sqrt{77284} = 278$

Answer: $278$

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

$\sqrt{8} \times \sqrt{18} = \sqrt{8 \times 18} = \sqrt{144}$

$= \sqrt{12 \times 12} = 12$

Answer: $12$

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

$\therefore \sqrt{17} \times \sqrt{17} = \sqrt{17^2} = 17$

Answer: $17$

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

$\displaystyle \frac{\sqrt{96}}{\sqrt{150}} = \sqrt{\frac{96}{150}} = \sqrt{\frac{6 \times 16}{6 \times 25}}$

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

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

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

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

Answer: $\sqrt{14}$

LHS $= \sqrt{121} + \sqrt{9} = 11 + 3 = 14 = \sqrt{196}$

RHS $= \sqrt{121 + 9} = \sqrt{130}$

$\therefore$ LHS $\ne$ RHS

Answer: LHS $= 14 = \sqrt{196}$, RHS $= \sqrt{130}$; LHS $\ne$ RHS

(i) LHS $= \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25$

RHS $= a + b = 7 + 24 = 31$

$25 \ne 31$, so $\sqrt{a^2 + b^2} \ne a + b$

(ii) LHS $= \sqrt{(7 + 24)^2} = \sqrt{31^2} = 31$

RHS $= 7 + 24 = 31$

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

Answer: (i) $25 \ne 31$ (ii) $31 = 31$

$\displaystyle \sqrt{0.0361} = \sqrt{\frac{361}{10000}}$

$\displaystyle = \frac{\sqrt{361}}{\sqrt{10000}}$

$\displaystyle = \frac{19}{100}$

$= 0.19$

Answer: $0.19$

$\displaystyle \sqrt{0.009025} = \sqrt{\frac{9025}{1000000}}$

$\displaystyle = \frac{\sqrt{9025}}{\sqrt{1000000}}$

$95^2 = 9025$, so $\sqrt{9025} = 95$

$\displaystyle = \frac{95}{1000}$

$= 0.095$

Answer: $0.095$

$\displaystyle \sqrt{86.49} = \sqrt{\frac{8649}{100}} = \frac{\sqrt{8649}}{\sqrt{100}}$

$\sqrt{8649}$ by the long division method:

Pair the digits from the right: $\overline{86}\;\overline{49}$

$9 \times 9 = 81 \le 86$: first digit $9$, and $86 - 81 = 5$

Bring down $49$: we get $549$. New divisor: $9 \times 2 = 18$; $183 \times 3 = 549$ ($184 \times 4 = 736$ is too big); $549 - 549 = 0$

$\sqrt{8649} = 93$

$\displaystyle \therefore \sqrt{86.49} = \frac{93}{10} = 9.3$

Answer: $9.3$