Tag: squares and square roots

Questions Related to squares and square roots

If $\sqrt{24}\, =\, 4.899,$ then the value of $\displaystyle \frac{8}{3}$ is

maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

  1. 0.544

  2. 2.666

  3. 1.633

  4. 1.333

Show answer
Correct Option: C
Explanation:

$\displaystyle {\sqrt{\frac{8}{3}} = \frac{\sqrt{8}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{24}}{3} = \frac{4.899}{3} = 1.633.}$ 

$\sqrt{1\, +\, \sqrt{1\, +\, \sqrt{1\, +\, .....}}}$ = ........

maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

  1. Equals $1$

  2. Lies between $0$ and $1$

  3. Lies between $1$ and $2$

  4. Is greater than $2$

Show answer
Correct Option: C
Explanation:

Let $x=\sqrt { 1+\sqrt { 1+\sqrt { 1+....... }  }  }$

$\therefore { x }^{ 2 }=1+\sqrt { 1+\sqrt { 1+\sqrt { 1+....... }  }  }$
$\Longrightarrow { x }^{ 2 }=1+x$
$\Longrightarrow { x }^{ 2 }-x-1=0$
$\Longrightarrow x=\cfrac { 1\pm \sqrt { 1+4 }  }{ 2 }$
$ \Longrightarrow x=\cfrac { 1\pm \sqrt { 5 }  }{ 2 }$
$ \Longrightarrow \cfrac { 1\pm 2.236 }{ 2 } =\left( -0.618, 1.618 \right) $
We reject the negative value because, from the given expression,

$x$ is positive.
So, $x=1.618$ approximately.
$\therefore 1<x<2$

$\sqrt{(12\, +\, \sqrt{12\, +\, \sqrt{12\, +\, ........}})}\, =\, ?$

maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

  1. 3

  2. 4

  3. 6

  4. Greater than 6

Show answer
Correct Option: B
Explanation:
Let $\sqrt{(12 + \sqrt{12 + \sqrt{12 + ........}})} = x$
Then, $\sqrt{12 + x} = x$
$ \Rightarrow 12 + x = x^2$
$\Rightarrow x^2 - x - 12 = 0$ 
$\Rightarrow (x - 4) (x + 3) = 0$
$\Rightarrow x = 4$       ...(neglecting $x = -3$)

Find the square root of each of the following correct to three places of decimal.
$17$
$1.7$
$2.5$
$\displaystyle\frac{7}{8}$

maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

  1. $4.153\;;\;1.304\;;\;1.581\;;\;0.935$

  2. $4.123\;;\;1.304\;;\;1.581\;;\;0.995$

  3. $4.123\;;\;1.304\;;\;1.581\;;\;0.935$

  4. $4.123\;;\;1.304\;;\;1.501\;;\;0.935$

Show answer
Correct Option: C
Explanation:

$17$

$4.123$
$4$$+4$ $17$$16$
$822$$+2$ $100$$81$
$8243$ $1900$$1644$
$25600$$24729$                 $871$

$\therefore$ The square root of $17=4.123$

$1.7$

$0$ $1.303$
$1$$+1$ $1.7$$1$
$23$$+3$ $70$$69$
$2603$ $10000$$7809$
$1191$

$\therefore$ The square root of $1.7=1.303$

$2.5$

$0$ $1.581$
$1$$+1$ $2.5$$1$
$25$$+5$ $150$$125$
$308$$+8$ $2500$$2464$
$3161$ $3600$$3161$                 $439$

$\therefore$ The square root of $2.5=1.581$

$\frac{7}{8}$

$8$ $70$$64$ $0.875$
$60$$56$
$40$$40$             $0$
$0.935$
$0$ $0.875$$0$
$9$$9$ $87$$81$
$183$$+3$ $650$$549$
$1865$ $10100$$9325$
$925$

$\therefore$ The square root of $\frac{7}{8}=0.935$

The simplest form of $\sqrt{864}$ is

maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

  1. $12\sqrt{5}$

  2. $12\sqrt{3}$

  3. $12\sqrt{6}$

  4. $6\sqrt{2}$

Show answer
Correct Option: C
Explanation:

$\sqrt{864}=\sqrt{2^2\times2^2\times3^2\times2\times3}=2\times2\times3\sqrt{6}=12\sqrt{6}$

Find the least number which must be subtracted from each of the following numbers so as to get a perfect square Also find the square root of the perfect square so obtained 
$(i) 402, (ii) 1989, (iii) 3250, (iv) 825, (v) 4000$

maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

  1. Least number which must be subtracted:$(i) 2, (ii) 22, (iii) 1, (iv) 21, (v) 52$

    Square root of the perfect square:
    $(i) 20, (ii) 34 ,(iii) 55, (iv) 26, (v) 67$

  2. Least number which must be subtracted:$(i) 2, (ii) 53, (iii) 1, (iv) 41, (v) 31$

    Square root of the perfect square:
    $(i) 20, (ii) 44, (iii) 57, (iv) 28, (v) 63$

  3. Least number which must be subtracted:$(i) 6, (ii) 22, (iii) 50, (iv) 31, (v) 40$

    Square root of the perfect square:
    $(i) 19, (ii) 41, (iii) 49 ,(iv) 27 ,(v) 65$

  4. Least number which must be subtracted:$(i) 8, (ii) 41, (iii) 12, (iv) 56, (v) 4$

    Square root of the perfect square:
    $(i) 19 ,(ii) 22, (iii) 37, (iv) 26, (v) 61$

Show answer
Correct Option: B
Explanation:

For $402$

Nearest perfect square is $400$
$\sqrt{400}$ $=$ $20$
$402$ $-$ $400$ $=$ $2$
For $1989$
Nearest perfect square is $1936$

$\sqrt{1936}$ $=$ $44$
$1936$ $-$ $1989$ $=$ $53$

For $3250$
Nearest perfect square is $3249$

$\sqrt{3249}$ $=$ $57$
$3250$ $-$ $3249$ $=$ $1$
For $825$
Nearest perfect square is $784$
$\sqrt{784}$ $=$ $28$
$825$ $-$ $784$ $=$ $41$
For $4000$
Nearest perfect square is $3969$
$\sqrt{3969}$ $=$ $63$
$4000$ $-$ $3969$ $=$ $31$
From this, Option B is correct answer.

Mr. Hansraj wants to find the least number of boxes to be added to get a perfect square. He already has $7924$ boxes with him. How many more boxes are required?

maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

  1. $819$

  2. $412$

  3. $419$

  4. $176$

Show answer
Correct Option: D
Explanation:

Find square root by long division method.

$\therefore \sqrt {7924}$ = $89.01$

Hence, the perfect square number smaller than $7924$ is
$89^2 = 7921$
The next perfect square no. is $90^2 = 8100$

So, $8100 - 7924 = 176$
Therefore, the man needs $176$ more boxes in order to get a perfect square number.
So, option D is correct.

$\displaystyle \sqrt { 4.8\times { 10 }^{ 9 } } $ is closest in value to

maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

  1. $2200$

  2. $70000$

  3. $220000$

  4. $7000000$

  5. $22000000$

Show answer
Correct Option: B
Explanation:

$\displaystyle \sqrt { 4.8\times { 10 }^{ 9 } } $

$=\sqrt{48\times10^{8}}$$\simeq$$7\times10^{4}=70000$
$7^{2}$$=$$49$
Hence Option B is correct.

What is an approximate value of $\sqrt{9805}$?

maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

  1. 98.56

  2. 97.23

  3. 99.05

  4. 100.34

Show answer
Correct Option: C
Explanation:

$99^2$ = 9801
$100^2$ = 10000
In between this two squares, 9805 is placed.
So the average of $\frac{99 + 100}{2}= 99.5$
Then, $99.5^2 = 9900.25$
So, $\sqrt{9805} \approx 99.05$

Estimate the square root of 500.

maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

  1. 22.5

  2. 20.3

  3. 21.4

  4. 23.6

Show answer
Correct Option: A
Explanation:

$22^2$ = 484
$23^2$ = 576
In between this two square numbers, 500 is placed.
So average of $\frac{22 + 23}{2}= 22.5$
Then, $22.5^2 = 506.25$
So, $\sqrt{500} \approx 22.5$