Tag: squares and square roots

Questions Related to squares and square roots

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

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 $127$
$127^2=16129$
$x=16161-16129$
$x=32$