Squares and Square Roots: MCQ test, Set 8

Which of these is equal to $63 \times 63$ ?
A $2 \times 63$ B $2^{63}$ C $63^3$ D $63^2$

Answer and solution

Answer: (D) $63^2$

$63$ is multiplied by itself $2$ times.

$\therefore 63 \times 63 = 63^2$
Which number below is not the square of a whole number?
A $289$ B $441$ C $451$ D $529$

Answer and solution

Answer: (C) $451$

$289 = 17^2$ , $441 = 21^2$ , $529 = 23^2$ , so they are perfect squares.

$451$ is not in the table of squares.
The next number in $529, 576, 625, \ldots$ is
A $729$ B $676$ C $675$ D $674$

Answer and solution

Answer: (B) $676$

These are $23^2, 24^2, 25^2$ .

Next is $26^2 = 676$
$50^2 =$
A $2500$ B $250$ C $100$ D $25000$

Answer and solution

Answer: (A) $2500$

$50^2 = 50 \times 50$

$5 \times 5 = 25$ , then write two zeros: $2500$
$105^2 = \square 25$ . The number in the box is
A $21$ B $100$ C $110$ D $121$

Answer and solution

Answer: (C) $110$

Rule: multiply the number before $5$ by the next number, then write $25$ .

$10 \times 11 = 110$ , so $105^2 = 11025$
If $161^2 = 25921$ , then $\sqrt{25921} =$
A $12960.5$ B $\sqrt{161}$ C $161$ D $322$

Answer and solution

Answer: (C) $161$

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

$161^2 = 25921 \Rightarrow \sqrt{25921} = 161$

(The square root is not half the number.)
$\sqrt{169} =$
A $84.5$ B $12$ C $14$ D $13$

Answer and solution

Answer: (D) $13$

$13^2 = 13 \times 13 = 169$

$\therefore \sqrt{169} = 13$
$\sqrt{\square} = 19$ . The number in the box is
A $361$ B $9.5$ C $\sqrt{19}$ D $38$

Answer and solution

Answer: (A) $361$

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

$19^2 = 19 \times 19 = 361$
$10816 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 13 \times 13$ . So $\sqrt{10816} =$
A $104$ B $26$ C $208$ D $5408$

Answer and solution

Answer: (A) $104$

Take one factor from each pair: $2 \times 2 \times 2 \times 13 = 104$
$\sqrt{2 \times 2 \times 7 \times 7 \times 11 \times 11} =$
A $77$ B $20$ C $23716$ D $154$

Answer and solution

Answer: (D) $154$

$\sqrt{2 \times 2 \times 7 \times 7 \times 11 \times 11} = \sqrt{(2 \times 2) \times (7 \times 7) \times (11 \times 11)}$

$= 2 \times 7 \times 11 = 154$
By the long division method, $\sqrt{2116} =$
A $46$ B $44$ C $56$ D $36$

Answer and solution

Answer: (A) $46$

Pairs: $\overline{21}\;\overline{16}$ .

$4 \times 4 = 16 \le 21$ : first digit $4$ , and $21 - 16 = 5$

Bring down $16$ : we get $516$ . New divisor: $4 \times 2 = 8$ ; $86 \times 6 = 516$ ( $87 \times 7 = 609$ is too big); $516 - 516 = 0$

Remainder $0$ . $\therefore \sqrt{2116} = 46$
While finding $\sqrt{3025}$ by long division, the first digit of the root is $5$ . The next divisor begins with
A $6$ B $25$ C $5$ D $10$

Answer and solution

Answer: (D) $10$

Double the root found so far: $5 \times 2 = 10$

Then $105 \times 5 = 525$ ( $106 \times 6 = 636$ is too big), so $\sqrt{3025} = 55$
The square root of $277729$ has how many digits?
A $2$ B $3$ C $4$ D $6$

Answer and solution

Answer: (B) $3$

Pairs from the right: $\overline{27}\;\overline{77}\;\overline{29}$ , that is $3$ groups.

One digit of the root for each group: $\sqrt{277729} = 527$
$\sqrt{8} \times \sqrt{32} =$
A $256$ B $128$ C $\sqrt{40}$ D $16$

Answer and solution

Answer: (D) $16$

$\sqrt{8} \times \sqrt{32} = \sqrt{8 \times 32} = \sqrt{256} = 16$
$(\sqrt{33})^2 =$
A $1089$ B $33$ C $\sqrt{33}$ D $66$

Answer and solution

Answer: (B) $33$

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

$\therefore (\sqrt{33})^2 = 33$
$\displaystyle \frac{\sqrt{90}}{\sqrt{10}} =$
A $3$ B $4.5$ C $\sqrt{80}$ D $9$

Answer and solution

Answer: (A) $3$

$\displaystyle \frac{\sqrt{90}}{\sqrt{10}} = \sqrt{\frac{90}{10}} = \sqrt{9} = 3$
$\displaystyle \frac{38}{\sqrt{38}} =$
A $38\sqrt{38}$ B $\sqrt{38}$ C $1$ D $38$

Answer and solution

Answer: (B) $\sqrt{38}$

$\displaystyle \frac{38}{\sqrt{38}} = \frac{\sqrt{38} \times \sqrt{38}}{\sqrt{38}} = \sqrt{38}$
$\sqrt{729} + \sqrt{1296} =$
A $2025$ B $45$ C $63$ D $3969$

Answer and solution

Answer: (C) $63$

$\sqrt{729} + \sqrt{1296} = 27 + 36 = 63$

$\sqrt{a} + \sqrt{b} \ne \sqrt{a + b}$ , so do not add first.
$\sqrt{400} - \sqrt{256} =$
A $4$ B $144$ C $12$ D $36$

Answer and solution

Answer: (A) $4$

$\sqrt{400} - \sqrt{256} = 20 - 16 = 4$

$\sqrt{a} - \sqrt{b} \ne \sqrt{a - b}$ , so do not subtract first.
$\sqrt{18^2 + 24^2} =$
A $900$ B $30$ C $42$ D $\sqrt{42}$

Answer and solution

Answer: (B) $30$

$\sqrt{18^2 + 24^2} = \sqrt{324 + 576} = \sqrt{900} = 30$

$\sqrt{a^2 + b^2} \ne a + b$
$\sqrt{(5 + 11)^2} =$
A $\sqrt{146}$ B $256$ C $16$ D $146$

Answer and solution

Answer: (C) $16$

$\sqrt{(a + b)^2} = a + b$ when $a + b > 0$

$\sqrt{(5 + 11)^2} = \sqrt{16^2} = 16$
$\sqrt{0.0064} =$
A $0.0032$ B $0.8$ C $0.008$ D $0.08$

Answer and solution

Answer: (D) $0.08$

$\displaystyle \sqrt{0.0064} = \sqrt{\frac{64}{10000}}$

$\displaystyle = \frac{\sqrt{64}}{\sqrt{10000}}$

$\displaystyle = \frac{8}{100}$

$= 0.08$
$\sqrt{0.000025} =$
A $0.05$ B $0.0005$ C $0.005$ D $0.0000125$

Answer and solution

Answer: (C) $0.005$

$\displaystyle \sqrt{0.000025} = \sqrt{\frac{25}{1000000}}$

$\displaystyle = \frac{\sqrt{25}}{\sqrt{1000000}}$

$\displaystyle = \frac{5}{1000}$

$= 0.005$
$\sqrt{1.96} =$
A $14$ B $1.4$ C $0.98$ D $0.14$

Answer and solution

Answer: (B) $1.4$

$\displaystyle \sqrt{1.96} = \sqrt{\frac{196}{100}} = \frac{\sqrt{196}}{\sqrt{100}} = \frac{14}{10} = 1.4$
$\sqrt{\square} = 2.2$ . The number in the box is
A $4.4$ B $4.84$ C $48.4$ D $0.484$

Answer and solution

Answer: (B) $4.84$

$\displaystyle 2.2 = \frac{22}{10}$

$\displaystyle \left(\frac{22}{10}\right)^2 = \frac{484}{100} = 4.84$

Check: $\displaystyle \sqrt{4.84} = \frac{\sqrt{484}}{\sqrt{100}} = 2.2$

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MCQ test, Set 8