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Estimating square roots - class-VIII

Description: estimating square roots
Number of Questions: 61
Created by:
Tags: squares and square roots square and square roots maths square roots and cube roots numbers squares, square roots, cubes, cube roots integer, power and roots
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Find the square root by guessing the units and tens digit:
$2116$
maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

  1. $46$

  2. $50$

  3. $52$

  4. $58$

Show answer
Correct Option: A
Explanation:

Its unit digit is 6 and tens digit is 1


For unit digit 6 , there are two possibilities either the square root end with 6 or 4

And according to options $46$ is the correct option

The number which exceeds its positive square root by $12$ is

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

  1. $9$

  2. $16$

  3. $25$

  4. None of these

Show answer
Correct Option: B
Explanation:
Let the positive number be x according to question,

$\sqrt{x}+12=x$

$\Rightarrow \sqrt{x}=x-12$

Squaring both sides,

$\Rightarrow x=x^{2}-24x+144$

$x^{2}-25x+144=0$

$x^{2}-16x-9x+144=0$

$x(x-16)-9(x-16)=0$

$(x-9)(x-16)=0$

 $  (x-9)=0 $ or $    (x-16)=0 $

 $ x=9 $  or $ x=16 $

So, the number is $16$.

Find the square root of which of the following numbers will be the least :

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

  1. $7\dfrac{58}{81}$

  2. $11\dfrac{14}{25}$

  3. $10\dfrac{1}{36}$

  4. $0.3481$

Show answer
Correct Option: D
Explanation:

$A.$


$7\dfrac{58}{81}=\dfrac{625}{81}$


$\Rightarrow$  $\sqrt{\dfrac{625}{81}}=\dfrac{25}{9}=2.77$

$B.$

$11\dfrac{14}{25}=\dfrac{289}{25}$

$\Rightarrow$  $\sqrt{\dfrac{289}{25}}=\dfrac{17}{5}=3.4$

$C.$

$10\dfrac{1}{36}=\dfrac{361}{36}$

$\Rightarrow$  $\sqrt{\dfrac{361}{36}}=\dfrac{19}{6}=3.16$

$D.$

$0.3481=\dfrac{3481}{10000}$

$\Rightarrow$  $\sqrt{\dfrac{3481}{10000}}=\dfrac{59}{100}=0.59$

$\therefore$  We can see, $0.3481$  has least square root.

Simlify: $\sqrt{\dfrac{-17}{144}-i}$

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

  1. $ \pm \left( {\dfrac{3}{2} - \dfrac{i}{3}} \right)$

  2. $ \pm \left( {\dfrac{3}{4} - \dfrac{{2i}}{3}} \right)$

  3. $ \pm \left( {\dfrac{3}{5} - \dfrac{{5i}}{6}} \right)$

  4. $ \pm \left( {\dfrac{2}{3} - \dfrac{{3i}}{4}} \right)$

Show answer
Correct Option: D
Explanation:

We know that,

${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$

 

Now, let,

$ -2ab=-i $

$ ab=i $

 

Now, consider $\dfrac{-17}{144}$. We can write it as,

$\dfrac{-17}{144}=\dfrac{64-81}{9\times 16}=\dfrac{4}{9}-\dfrac{9}{16}$

 

Thus,

$ \sqrt{\dfrac{-17}{144}-i}=\sqrt{\dfrac{4}{9}-\dfrac{9}{16}-i} $

$ =\sqrt{{{\left( \dfrac{2}{3} \right)}^{2}}+{{\left( i \right)}^{2}}{{\left( \dfrac{3}{4} \right)}^{2}}-2\times \left( \dfrac{2}{3} \right)\times \left( \dfrac{3i}{4} \right)} $

$ =\sqrt{{{\left( \dfrac{2}{3}-\dfrac{3i}{4} \right)}^{2}}} $

$ =\pm \left( \dfrac{2}{3}-\dfrac{3i}{4} \right) $

 

Hence, this is the required result.

The approximate value of$\sqrt { { \left( 1.97 \right)  }^{ 2 }{ \left( 4.02 \right)  }^{ 2 }{ \left( 3.98 \right)  }^{ 2 } }$

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

  1. $31.59 $

  2. $5.099$

  3. $5.009$

  4. $5.734$

Show answer
Correct Option: A
Explanation:

We have,

$ \sqrt{{{\left( 1.97 \right)}^{2}}{{\left( 4.02 \right)}^{2}}{{\left( 3.98 \right)}^{2}}} $

$ =1.97\times 4.02\times 3.98 $

$ =31.59 $

Hence, this is the answer.

Find the square root correct upto $2$ decimals $60.92$.

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

  1. $7.92$

  2. $7.81$

  3. $7.27$

  4. $7.56$

Show answer
Correct Option: B
Explanation:
$\sqrt{6092}$

$=\sqrt{\dfrac{6092}{100}}$

$=\dfrac{78.05}{10}$

$=7.81$

The positive square root of $( \sqrt { 48 } - \sqrt { 45 } )$ is _________.

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

  1. $\frac { \sqrt [ 4 ] { 3 } } { \sqrt { 2 } } ( \sqrt { 5 } - \sqrt { 3 } )$

  2. $\frac { \sqrt [ 4 ] { 3 } } { 2 } ( \sqrt { 5 } - \sqrt { 3 } )$

  3. $\frac { \sqrt { 2 } } { \sqrt [ 4 ] { 3 } } ( \sqrt { 5 } - \sqrt { 3 } )$

  4. $\frac { \sqrt [ 4 ] { 3 } } { \sqrt { 2 } } ( \sqrt { 5 } + \sqrt { 3 } )$

Show answer
Correct Option: A
Explanation:

Solving $(\sqrt{48}-\sqrt{45})$

$4\sqrt{3}-3\sqrt{5}$

$\dfrac{\sqrt{3}}{2}(8-2\sqrt{15})$

$\dfrac{\sqrt{3}}{2}(3+5-2\sqrt{15})$

$\dfrac{\sqrt{3}}{2}(\sqrt{3^2}+\sqrt{5^2}-2\sqrt{5}\times\sqrt{3})$

$\dfrac{\sqrt3}{2}(\sqrt5-\sqrt3)^2$

$Now \ Finding\ Square \ Root$

$\pm{ \dfrac{3^{\frac{1}{4}}}{\sqrt2}(\sqrt5-\sqrt3)}$

$So \ it's \ positive \ root \ is \ $$ \dfrac{3^{\frac{1}{4}}}{\sqrt2}(\sqrt5-\sqrt3)$
Correct Answer is $A$

If $\sqrt{2}=1.414$ then the value of $\sqrt{8}$ is

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

  1. $2.828$

  2. $1.828$

  3. $2.282$

  4. $2.288$

Show answer
Correct Option: A
Explanation:
$\sqrt{2}=1.414$(given)
$\sqrt{8}=\sqrt{2\times 2\times 2}=2\sqrt{2}=2\times 1.414=2.828$
$\therefore \sqrt{8}=2.828$

Find the square root of 
$5-2\sqrt{6}$

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

  1. $\sqrt{13}-\sqrt{2}$

  2. $\sqrt{3}-\sqrt{2}$

  3. $\sqrt{5}-\sqrt{3}$

  4. $\sqrt{5}-\sqrt{2}$

Show answer
Correct Option: B
Explanation:

$5-2\sqrt{6}=3+2-2\sqrt{6}$


$=({\sqrt{3}})^2+({\sqrt{2}})^2-2\sqrt{3}\times \sqrt {2}$

Using $(a-b)^2=a^2+b^2-2ab$


${5-2\sqrt{6}}=(\sqrt{3}-\sqrt{2})^2$

$\sqrt {5-2\sqrt{6}}=(\sqrt{3}-\sqrt{2})$

$So, \  the \  square \  root \  of \  (5-2\sqrt{6})=\sqrt{3}-\sqrt{2}$

$\sqrt{(a - b)^2} + \sqrt{(b - a)^2}$ is

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

  1. Always zero

  2. Never zero

  3. Positive if and only if a > b

  4. Positive only if a $\ne$ b

Show answer
Correct Option: D
Explanation:

$\sqrt{(a - b)^2} + \sqrt{(b - a)^2}$
$= |a - b| + |b - a|$

Now, If $a > b$
$= a - b + a - b$
$= 2a - 2b$...+ ve

If $b > a$
$= b - a + b - a$
$= 2b - 2a$...+ ve
Therefore, if $a \ne b$ then the given equation is always positive.
Hence, option 'D' is correct.

The value of $\displaystyle \frac{1\, +\, \sqrt{0.01}}{1\, -\, \sqrt{0.1}}$ is close to .......... .

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

  1. 0.6

  2. 1.1

  3. 1.6

  4. 1.7

Show answer
Correct Option: C
Explanation:

$\displaystyle \frac{1\, +\, \sqrt{0.01}}{1\, -\, \sqrt{0.1}}\, =\, \displaystyle \frac{1\, +\, 0.1}{1\, -\, 0.32}\, =\, \displaystyle \frac{1.1}{0.68}\, =\, 1.6$

If $\sqrt{6}\, =\, 2.55,$ then the value of $\displaystyle {\sqrt{\frac{2}{3}\, +\, 3\frac{3}{2}}}$ is

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

  1. 4.48

  2. 4.49

  3. 4.50

  4. None of these

Show answer
Correct Option: D
Explanation:
$ {\sqrt{\cfrac{2}{3} + 3\cfrac{3}{2}}}$
$=  {\cfrac{\sqrt{2}}{\sqrt{3}} \times \cfrac{\sqrt{3}}{\sqrt{3}} + 3 \times \cfrac{\sqrt{3}}{\sqrt{2}} \times \cfrac{\sqrt{2}}{\sqrt{2}}}$
$=  {\cfrac{\sqrt{6}}{3} + \cfrac{3\sqrt{6}}{2} = \cfrac{2.55}{3} + \cfrac{3 \times 2.55}{2}}$
$=  {\cfrac{2.55}{3} + \cfrac{7.65}{2} = \cfrac{5.10 + 22.95}{6}}$
$=  \cfrac{28.05}{6} = 4.675$

$\displaystyle {\sqrt{\frac{4}{3}}\, -\, \sqrt{\frac{3}{4}}\, =\, ?}$

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

  1. $\displaystyle \frac{1}{2\sqrt{3}}$

  2. $\displaystyle - \frac{1}{2\sqrt{3}}$

  3. 1

  4. $\displaystyle \frac{5\sqrt{3}}{6}$

Show answer
Correct Option: A
Explanation:

$\displaystyle {\frac{\sqrt{4}}{\sqrt{3}} - \frac{\sqrt{3}}{\sqrt{4}} = \frac{2}{\sqrt{3}} - \frac{\sqrt{3}}{2} = \frac{4 - 3}{2\sqrt{3}} = \frac{1}{2\sqrt{3}}}$

If $\sqrt{75.24\, +\, x}\, =\, 8.71,$ then the value of x is

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

  1. 0.6241

  2. 6.241

  3. 62.41

  4. None of these

Show answer
Correct Option: A
Explanation:
$\sqrt{75.24\, +\, x}\, =\, 8.71$        ...Given
Squaring on both sides, we get
$75.24 + x = 8.71 \times 8.71$
$\Rightarrow x=8.71^2-75.24$
$ \Rightarrow x = 0.6241$

If $\sqrt{3}\, =\, 1.732,$ then the approximate value of $\displaystyle \frac{1}{\sqrt{3}}$ is

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

  1. 0.617

  2. 0.313

  3. 0.577

  4. 0.173

Show answer
Correct Option: C
Explanation:

$ {\cfrac{1}{\sqrt{3}} = \cfrac{1}{\sqrt{3}} \times \cfrac{\sqrt{1}}{\sqrt{3}} = \cfrac{\sqrt{1}}{3} = \cfrac{1.732}{3} = 0.577}$

$\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}}\, =\, ?$

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

  1. 0

  2. 1

  3. 2

  4. $2^{31/32}$

Show answer
Correct Option: D
Explanation:

$\sqrt{2\, \times\, \sqrt{2\, \times\, \sqrt{2\, \times\, \sqrt{2\, \times\, 2^{1/2}}}}}$

$=\, \sqrt{2\, \times\, \sqrt{2\, \times\, \sqrt{(2\, \times\, 2^{3/4})}}}$
$=\, \sqrt{2\, \times\, \sqrt{2\, \times\, 2^{7/8}}}\, =\, \sqrt{2\, \times\, 2^{15/16}}\, =\, 2^{31/32}$

By using the table for square root find the value of $\sqrt{7}$.

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

  1. $2.652$

  2. $2.746$

  3. $2.646$

  4. $2.616$

Show answer
Correct Option: C
Explanation:

The sqaure root of $\sqrt 7$ is $2.646$

By using the table for square root find the value of
$13.21$
$21.97$

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

  1. 3.63, 4.60

  2. 3.63, 4.69

  3. 3.53, 4.69

  4. 3.63, 4.19

Show answer
Correct Option: B
Explanation:

(i)

From square root table, Square root of 13.21 is:

 √13.21 = 3.6345

Therefore,

The square root of 13.21 is 3.63

(ii) From square root table, Square root of 21.97 is:

 √21.97 = 4.687

Therefore,

The square root of 21.97 is 4.69

Find the square root of $10$, correct to four places of decimal.

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

  1. 3.4623

  2. 3.1023

  3. 3.1693

  4. 3.1623

Show answer
Correct Option: D
Explanation:
$3.16227$
$3$$+3$ $10$$9$
$61$$+1$ $100$$61$
$626$$+6$ $3900$$3756$
$6322$$+2$ $14400$$12644$
$63242$$+2$-------------$632447$ $175600$$126484$---------------$4911600$$4427129$

$\sqrt{10}=3.16227\simeq 3.1623$
$\therefore$ The square root of $10$ correct to four places of decimal is $3.1623$

The square root of $\displaystyle \frac{\left ( 3\frac{1}{4} \right )^{4}-\left ( 4\frac{1}{3} \right )^{4}}{\left ( 3\frac{1}{4} \right )^{2}-\left ( 4\frac{1}{3} \right )^{2}}$ is

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

  1. $\displaystyle 7\frac{5}{12}$

  2. $\displaystyle 7\frac{7}{12}$

  3. $\displaystyle 5\frac{5}{12}$

  4. $\displaystyle 5\frac{7}{12}$

Show answer
Correct Option: C
Explanation:

$\frac{\left ( 3\tfrac{1}{4} \right )^{4}-\left ( 4\tfrac{1}{3} \right )^{4}}{\left ( 3\tfrac{1}{4} \right )^{2}-\left ( 4\tfrac{1}{3} \right )^{2}}$

=$\frac{\left [ \left ( 3\tfrac{1}{4} \right )^{2}+\left ( 4\tfrac{1}{3} \right )^{2} \right ]\left [ \left ( 3\tfrac{1}{4} \right )^{2}-\left ( 4\tfrac{1}{3} \right )^{2} \right ]}{\left ( 3\tfrac{1}{4} \right )^{2}-\left ( 4\tfrac{1}{3} \right )^{2}}$
=$\left ( 3\tfrac{1}{4} \right )^{2}+\left ( 4\tfrac{1}{3} \right )^{2}$
=$\left ( \frac{13}{16} \right )^{2}+\left ( \frac{13}{9} \right )^{2}=169\times \left ( \frac{9+16}{144} \right )=169\times\frac{25}{144}$ 
Then squire root =$\frac{13\times 5}{12}=\frac{65}{12}$=$5\frac{5}{12}$

Estimate: $\sqrt { 60 } $

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

  1. $7.7$

  2. $7$

  3. $7.2$

  4. $7.5$

Show answer
Correct Option: A
Explanation:

$60$ is in between two perfect squares: $49$, which is ${7}^{2}$ and $64$ which is ${8}^{2}$. The difference between $64$ and $49$ is $15$ so $60$ is little more than $\cfrac{2}{3}$ of the way toward $64$ from $49$. A reasonable estimate for $\sqrt {60}$, then would be about $7.7$ which is a little more than $\cfrac{2}{3}$ toward $8$ from $7$.

If $\sqrt{0.01+\sqrt{0.0064}}=x$, then the value of $x$ is ____________.

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

  1. $0.3$

  2. $0.03$

  3. $\sqrt{0.18}$

  4. None of these

Show answer
Correct Option: A
Explanation:

Given, $\sqrt{0.01+\sqrt{0.0064}}=x$

Taking square of both sides.
$0.01+\sqrt{0.0064}=x^2$
We know that $\sqrt{0.0064}=0.08$
So, $x^2=0.08+0.01$
$x^2=0.09$
$x=0.3$

The square root of $\displaystyle\frac{36}{5}$ correct to two decimal places is _____________.

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

  1. $2.68$

  2. $2.69$

  3. $2.67$

  4. $2.66$

Show answer
Correct Option: A
Explanation:

We know $\sqrt{\dfrac{36}{5}}=\dfrac {6}{\sqrt 5}$

$=\dfrac{6}{2.236}$
$=2.6833$
Upto two decimal places:
$=2.68$

Which of the following is not a perfect square?

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

  1. $16384$

  2. $23857$

  3. $18496$

  4. $11025$

Show answer
Correct Option: B
Explanation:

$16384= 128\times 128$         (perfect square)

$18496=136\times 136$           (perfect square)
$11025=105\times 105$         (perfect square)
$23857 = 1\times  23857$        (prime no. so not a perfect square)
Hence, option B is correct.

The number that must be subtracted from $16161$ to get a perfect square is ________.

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

  1. $31$

  2. $32$

  3. $33$

  4. $34$

Show answer
Correct Option: B
Explanation:

Let the number is $x$

Finding the square root of $16161$
$\sqrt{16161}=127.1259$

Finding the square of $127$
$127^2=16129$
$x=16161-16129$
$x=32$

Determine
(a) $\sqrt{147} \times \sqrt{243} \$

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

  1. $189$

  2. $209$

  3. $197$

  4. $138$

Show answer
Correct Option: A
Explanation:

$√143 = √3  × √49 = √3  × 7$
$√243 = √3  × √81 = √3  × 9$
$√143  ×√243  = 3  × 9  × 7 = 189$

If $x=\sqrt {12}-\sqrt {9},y=\sqrt {13}-\sqrt {10}$ and $z=\sqrt {11}-\sqrt {8}$, then which of the following is true?

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

  1. $z > x > y$

  2. $z > y > x$

  3. $y > x > z$

  4. $y > z> x$

Show answer
Correct Option: A
Explanation:

$x=\sqrt{12}-\sqrt{9}$

$x=2\sqrt{3}-3$
$x=2(1.73)-3$
$x=3.46-3$
$\therefore$   $x=0.46$                    ----- ( 1 )
Now,
$y=\sqrt{13}-\sqrt{10}$
$y=3.60-3.16$
$\therefore$  $y=0.44$                ---- ( 2 )
Now,
$z=\sqrt{11}-\sqrt{8}$
$z=3.32-2.83$
$\therefore$  $z=0.49$             ---- ( 3 )
From ( 1 ), ( 2 ) and ( 3 )
$\Rightarrow$  $0.49>0.46>0.44$
$\therefore$  $z>x>y$

Find the atleast number which must  be added to each of the following numbers to get a perfect square. Also find the square root of the perfect square numbers.


$a)525$

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

  1. 4

  2. 3

  3. 1

  4. 6

  5. 12

Show answer
Correct Option: A
Explanation:
$i)\ 23^2=529$
$\therefore \ $ it will add $4$ to $525$ we get $529$
which is payout square
$\therefore \ 525+4=529=23^2$

$\therefore 4$ should be added


The value of $\sqrt { \sqrt [ a ]{ { 4 }^{ { a }^{ { a }^{ 2 } } }\sqrt { { 6 }^{ { a }^{ 3 } }\sqrt [ { a }^{ 3 } ]{ { 12 }^{ { a }^{ 6 } }\sqrt [ { a }^{ 4 } ]{ { 18 }^{ { a }^{ 10 } } }  }  }  }  } $ is equal to

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

  1. $\sqrt {216}$

  2. $\sqrt {72}$

  3. $72$

  4. $216$

Show answer
Correct Option: C
Explanation:

$\begin{array}{l} =\sqrt { \sqrt [ a ]{ { { 4^{ { a^{ { a^{ 2 } } } } } }\sqrt { { 6^{ { a^{ 3 } } } }\sqrt [ { { a^{ 3 } } } ]{ { { { 12 }^{ { a^{ 6 } } } }\sqrt [ { { a^{ 4 } } } ]{ { { { 18 }^{ { a^{ 10 } } } } } }  } }  }  } }  }  \ =\sqrt { 4\times 6\times 18\times 12 }  \ =72 \ Hence, \ option\, \, C\, \, is\, correct\, \, answer. \end{array}$

The approximate value of $\sqrt{(1.97)^2+(4.02)^2+(3.98)^2}$.

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

  1. $5.99$

  2. $5.099$

  3. $5.009$

  4. $5.734$

Show answer
Correct Option: A
Explanation:

According to Question

$\sqrt{(2-0.03)^2+(4+0.02)^2+(4-0.02)^2}$
Opening Brackets
$\sqrt{(4+0.0009-0.12)+(16+0.0004+0.16)+(16+0.0004-0.16)}$
$\sqrt{36-0.1183}$
$\sqrt{35.8817} \approx 5.99$

The square root of 496 correct to three places of decimal is 22.271
State true or false.

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

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

$\quad \ \quad \quad \quad \quad \quad 22.271\ \quad \quad \quad \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ \quad \quad 2)496.00\quad 00\quad 00\ \quad \quad \quad -4\quad \downarrow \ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ 42)\quad 00\quad 96\ \quad \quad -0084\quad \quad \downarrow \ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ 442)\quad 12\quad 00\ \quad \quad -\quad \quad 0884\quad \quad \downarrow \ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \quad \ 4447)\quad \quad \quad \quad \quad \quad 316\quad 00\quad \ \quad \quad \quad \quad \quad \quad \quad \quad \quad -31129\quad \quad \downarrow \ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ 44541)\qquad \qquad \qquad \quad \quad \quad \quad 471\quad 00\ \qquad \qquad \qquad \qquad \qquad -\quad \quad 44541\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 2559\ Rounding\quad off\quad :\quad 22.271\ \ $

Find the square root of $\displaystyle 3 \frac{4}{5}$  to two decimal places 

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

  1. $1.95$

  2. $1.96$

  3. $1.84$

  4. $1.94$

Show answer
Correct Option: A
Explanation:

  $\displaystyle 3 \dfrac{4}{5}$
$=\dfrac{19}{5}$
$=3.8$

$1$$1$ $3.8$$1$
$29$$9$ $280$$261$
$384$$4$ $1900$$1536$
$3889$ $36400$$35001$                     

Therefore,
Square root of $\displaystyle 3 \frac{4}{5}=1.949$
                              $=1.95$

State true or false:

The square root of 0.008 correct to three decimal places is 0.089.

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

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

$\quad \ \quad \quad \quad \quad \quad 0.089\ \quad \quad \quad \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ \quad \quad 0)\quad 0.00\quad 80\quad 00\ \quad \quad \quad -0.00\quad \downarrow \ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ 8)\quad \quad 00\quad 80\ \quad \quad -\quad 0064\quad \quad \downarrow \ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ 169)\quad \quad \quad 16\quad 00\ \quad \quad -\quad \quad \quad \quad 1529\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \quad \ \quad \quad \quad \quad \quad \quad \quad \quad 71\quad \quad \quad \ Rounding\quad off\quad :\quad 0.089\ \ $

State true or false:

The square root of 5.2005 correct to two decimal places is 2.28.

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

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

$\quad \ \quad \quad \quad \quad \quad 2.287\ \quad \quad \quad \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ \quad \quad 2)5.\quad 20\quad 05\ \quad \quad \quad -4\quad \downarrow \ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ 42)01\quad 20\ \quad \quad -0084\quad \quad \downarrow \ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ 448)\quad 036\quad 05\ \quad \quad -\quad \quad 3584\quad \quad \downarrow \ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ \quad \quad \quad \quad \quad \quad \quad 21\quad $

The square root of 245 correct to two places of decimal is 15.65
State true or false

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

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

$\quad \ \quad \quad \quad \quad \quad 15.65\ \quad \quad \quad \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ \quad 15)245.00\quad 00\ \quad \quad -225\quad \downarrow \ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ 306)\quad 20\quad 00\ \quad \quad -1836\quad \quad \downarrow \ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ 3125)164\quad 00\ \quad \quad -\quad \quad 15625\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \quad \ \quad \quad \quad \quad \quad \quad 845\quad \quad \ Rounding\quad off\quad :\quad 15.65\ \ $

The square root of 0.065 correct to three places of decimal is 0.255
State true or false

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

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

$\quad \ \quad \quad \quad \quad \quad 0.254\ \quad \quad \quad \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ \quad \quad 2)\quad 0.06\quad 50\quad 00\ \quad \quad \quad -0.04\quad \downarrow \ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ \quad 45)\quad \quad \quad 2\quad 50\ \quad \quad -\quad \quad 225\quad \quad \downarrow \ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ \quad 504)\quad \quad \quad 25\quad 00\ \quad \quad -\quad \quad \quad \quad 2016\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \quad \ \quad \quad \quad \quad \quad \quad \quad \quad 484\quad \quad \quad \ \ $

The square root of 82.6 correct to two places of decimal is 9.09
State true or false.

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

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

$\quad \ \quad \quad \quad \quad \quad 9.087\ \quad \quad \quad \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ \quad \quad 9)82.60\quad 00\ \quad \quad \quad -81\quad \downarrow \ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ 180)\quad 01\quad 60\ \quad \quad -0000\quad \quad \downarrow \ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ 1808)\quad 160\quad 00\ \quad \quad -\quad \quad 12864\quad \quad \downarrow \ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \quad \ 19167)\quad \quad \quad \quad \quad 3136\quad 00\quad \ \quad \quad \quad \quad \quad \quad \quad \quad \quad -31129\quad \quad \downarrow \ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ \qquad \qquad \qquad \qquad 282471\quad $

State true or false.
The square root of 0.602 correct to two decimal places is 0.78.

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

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

$\quad \ \quad \quad \quad \quad \quad 0.77\ \quad \quad \quad \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ \quad \quad 7)\quad 0.60\quad 20\ \quad \quad \quad -0.49\quad \downarrow \ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ 147)\quad \quad 11\quad 20\ \quad \quad -\quad 1029\quad \quad \downarrow \ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ \quad \quad \quad \quad \quad 191\quad $

Find the square root of $\displaystyle 1 \frac{5}{16}$  correct to two decimal places

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

  1. $1.14$

  2. $1.15$

  3. $1.65$

  4. $1.12$

Show answer
Correct Option: B
Explanation:

 $\displaystyle 1 \dfrac{15}{16}$

$=\dfrac{21}{16}$
$=1.3125$

$1$$1$ $1.3125$$1$
$21$  $1$ $31$$21$
$224$   $4$ $1025$$896$
$2285$ $12900$$11425$                  $1475$

Therefore,
Square root of $\displaystyle 1 \frac{15}{16}=1.145$
                                   $=1.15$

Find the square root of $\displaystyle 6 \frac{7}{8}$ correct to two decimal places 

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

  1. $2.62$

  2. $2.61$

  3. $2.63$

  4. $2.6$

Show answer
Correct Option: A
Explanation:

 $\displaystyle 6 \frac{7}{8}$
$=\dfrac{55}{8}$
$=6.875$

$2$$2$ $6.875$$4$
$46$  $6$ $287$$276$
$522$    $1150$$1044$
$6$                  

Therefore,
Square root of $\displaystyle 6 \frac{7}{8}=2.62$
                             

Consider the Following values of the three given number $\displaystyle \sqrt{103},$ $\displaystyle \sqrt{99.35},$ $\displaystyle \sqrt{102.20},$
1.10.1489 (approx,)
2.10.109(approx,)
3.9.967 (approx,)
The correct sequence of the these values matching with the above number is:

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

  1. 1, 2, 3

  2. 1, 3, 2

  3. 2, 3, 1

  4. 3, 1, 2

Show answer
Correct Option: B
Explanation:

$\sqrt{3}=10.14889$
$\sqrt{99.35}=9.967$
$\sqrt{102.30}=10.1094$
Hence,The correct sequence of the these values matching with the above number is:1,3,2

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

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 


$x=\sqrt{1+\sqrt{1-----}}$

Squaring both sides

$x^2=1+\sqrt{1+\sqrt{1+\sqrt{1------}}}$

$x^2=1+x$                $(\because x=\sqrt{1+\sqrt{1------}})$

$x^2-x-1=0$

finding roots, we get

$\dfrac{1\pm\sqrt{1+4}}{2}$

$=\dfrac{1\pm\sqrt{5}}{2}$

$=-0.615$ and $1.615$

If $x\, \ast\, y\, =\, \sqrt{x^2\, +\, y^2}$, then the value of $(1^{\ast}\, 2\, \sqrt{2})(1^{\ast}\, - 2\, \sqrt{2})$ is:  

maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares
    • 7

  2. 0

  3. 2

  4. 9

Show answer
Correct Option: D
Explanation:

$1^{\ast} 2 \sqrt{2}\, =\, \sqrt{(1)^2\, +\, (2 \sqrt{2}^2}\, =\, \sqrt{1\, +\, 8}\, =\, 3$

$1^{\ast} -2 \sqrt{2}\, =\, \sqrt{(1)^2\, +\, (-2 \sqrt)^2}\, =\, \sqrt{1\, +\, 8}\, =\, 3$
$(1\, \ast\, 2 \sqrt{2})(1\, \ast\, -2 \sqrt{2})\, =\, (3)(3)\, =\, 9$

$\displaystyle \frac{\sqrt{32}\, +\, \sqrt{48}}{\sqrt{8}\, +\, \sqrt{12}}\, =\, ?$

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

  1. $\sqrt{2}$

  2. 2

  3. 4

  4. 8

Show answer
Correct Option: B
Explanation:
$ {\cfrac{\sqrt{32} + \sqrt{48}}{\sqrt{8} + \sqrt{12}} = \cfrac{\sqrt{16 \times 2} + \sqrt{16 \times 3}}{\sqrt{4 \times 2} + \sqrt{4 \times 3}}}$

$= \cfrac{4\sqrt{2} + 4\sqrt{3}}{2\sqrt{2} + 2\sqrt{3}}$

$ = \cfrac{4 \left (\sqrt{2} + \sqrt{3} \right )}{2 \left (\sqrt{2} + \sqrt{3} \right )}$

$ = 2$

If $\sqrt{2}\, =\, 1.4142,$ then the value of $\displaystyle \frac{2}{9}$ is

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

  1. 0.2321

  2. 0.4714

  3. 0.3174

  4. 0.4174

Show answer
Correct Option: B
Explanation:

$\displaystyle {\sqrt {\frac{2}{9}}\, =\, \frac{\sqrt{2}}{\sqrt{9}}\, =\, \frac{\sqrt{2}}{3}\, =\, \frac{1.4142}{3}\, =\, 0.4714.}$

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, $\sqrt{500} \approx 22.5$

Find the approximate value of $\sqrt{5245}$.

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

  1. 70.5

  2. 72.3

  3. 71.8

  4. 79.2

Show answer
Correct Option: B
Explanation:

$72^2$ = 5184
$73^2$ = 5329
In between this two squares, 5245 is placed.
So average of $\frac{72 + 73}{2}= 72.5$
Then, $72.5^2 = 5256.25$
So, $\sqrt{5245} \approx 72.3$

Find the approximate value of $\sqrt{1235}$.

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

  1. $35.15$

  2. $32.19$

  3. $30.25$

  4. $29.13$

Show answer
Correct Option: A
Explanation:

$35^2$ = 1225
$36^2$ = 1296
In between this two squares, 1235 is placed.
So the average of $\dfrac{35 + 36}{2}= 35.5$
Then, $35.5^2 = 1260.25$

So $\sqrt{1235}\approx35.15$

Estimate the value of $\sqrt{750}$.

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

  1. 24.3

  2. 25.1

  3. 23.2

  4. 27.3

Show answer
Correct Option: D
Explanation:

$27^2$ = 729
$28^2$ = 784
In between this two squares, 750 is placed.
So average of $\frac{27 + 28}{2}= 27.5$
Then, $27.5^2 = 756.25$
So, $\sqrt{750} \approx 27.3$

Estimate the square root of $300$

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

  1. $12.44$

  2. $16.66$

  3. $17.32$

  4. $18.54$

Show answer
Correct Option: C
Explanation:

The square root of $300$ is $10\sqrt 3$

We know that $\sqrt 3=1.732$
Thus $\sqrt{300}=10\times 1.732=17.32$

Estiamate the square root of $850$ 

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

  1. $29.15$

  2. $30.21$

  3. $98.23$

  4. $23.11$

Show answer
Correct Option: A
Explanation:

The square root of $850$ is $\sqrt {850}=\sqrt {25 \times 34}=5\sqrt {34}$

Square root of $34$ lie between $5$ and $6$.
 square of $5.5$ is $30.25$ 
Now, we can say that square root of $34$ lie between $5.5$ and $6$.
Now, square of $5.75$ is $33.06$
So, square root of $34$ lie between $5.75$ and $6$.
Now, we have to choose the number $5.85$
$(5.85)^2=34.225$ which is greater than $34$ and close to $34$
So, assume a number $5.84$.
$(5.84)^2=34.1056$
$(5.83)^2=33.9889$
Hence, we can say that square root of 34 lie between $5.83$ and $5.84$.
So, $5\times 5.83=29.15$.

The real number $(\sqrt [3]{\sqrt {75} - \sqrt {12}})^{-2}$ when expressed in the simplest form is equal to

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

  1. $\dfrac {1}{2}$

  2. $\dfrac {1}{3}$

  3. $\dfrac {1}{4}$

  4. $\dfrac {1}{5}$

Show answer
Correct Option: B
Explanation:

Real number $(\sqrt[3]{\sqrt{75}-\sqrt{12}})^{-2}$

$\sqrt{75}=5\sqrt{3}$ and $\sqrt{12}=2\sqrt{3}$
$=(\sqrt[3]{5\sqrt{3}-2\sqrt{3}})^{-2}$
$=(\sqrt[3]{3\sqrt{3}})^{-2}$
$(3\sqrt{3})^{\dfrac{-2}{3}} ....... (1)$
$3\sqrt{3}=3^{\dfrac{1}{2}+1}=3^{\dfrac{3}{2}} ....... (ii)$
Substituting $(ii)$ in $(i)$
$\left[(3)^{\dfrac{3}{2}}\right]^{\dfrac{-2}{3}}$
$=\dfrac{1}{3}=(3)^{-1}$

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