Square and Square root class 8

Squares and square root Both ideas are inherently opposed to one another. Squares are the results of multiplying a value by itself. In contrast, the square root of an integer is a value that, when multiplied by itself, returns the original value.
In the post “Square and Square root class 8″ we have given various type of problems based on square and square root for class 8 students. These questions are essential in building a strong foundation in mathematics and are often included in primary school level examinations.

Square and Square root class 8(Solved Examples)

1)Is 196 is a perfect square ?
Solution: 196= 2 x 2 x 7 x 7 = $\overline{2 \times 2} \times \overline{7 \times 7}$
$\because$

The prime factors of 196 can be grouped in pairs ; 196 is a perfect square .

2) Is 180 is a perfect square ?
Solution: 180= 2 x 2 x 3 x 3 x 5 = $\overline{2 \times 2} \times \overline{3 \times 3} \times 5$
$\because$ The prime factors of 180 can not be grouped in pairs ; 180 is a not perfect square .

3)Find the square root of the following numbers .
(i) 7056
Solution: 7056 =2 x 2 x 2 x 2 x 3 x 3 x 7 x 7
= $2^{2}\times 2^{2}\times 3^{2}\times 7^{2}$
$\Rightarrow \sqrt{7056} = 2 \times 2\times 3\times 7$
= 84

(ii) 42.25
Solution: $42.25=\frac{4225}{100}$ = $\frac{169}{4}$
$=\frac{13^{2}}{2^{2}}$

$\Rightarrow \sqrt{42.25}=\frac{13}{2}=6.5$

4) In an auditorium, the number of rows is equal to the number of chairs in each row . If the capacity of auditorium is 1764 ,find the number of chairs in each row.
Solution: Let the number of rows be x .
Then the number of chairs in each row = x
The total number of chairs in the auditorium = $x \times x = x^{2}$
But the capacity of the auditorium = 1764
$\therefore x^{2}=1764$
=2 x 2 x 3 x 3 x 7 x7
= $2^{2}\times 3^{2}\times 7^{2}$
$\Rightarrow x= 2\times 3\times 7$
= 42
Hence, the number of chairs in each row = 42

5) Find the value $\left ( 1\frac{3}{2} \right )^{2}$ .
Solution : $\left ( 1\frac{3}{2} \right )^{2}$ = $\frac{5}{2} \times \frac{5}{2}$

= $\frac{25}{4}$

= $6\frac{1}{4}$

6) What is the smallest number by which 3528 should be divided to make is a perfect square ?
Solution : 3528 = 2 x 2 x 42
If 3528 is divided by 2 , then 1764 = 42 x 42 , which is a perfect square.

7) Find the square root of 6.76 .
Solution : $\sqrt{6.76} =\sqrt{\frac{676}{100}}$

= $\sqrt{\frac{2\times 2\times 13\times 13 }{2\times 2\times 5\times 5}}$

= $\sqrt{\frac{\overline{2\times 2}\times\overline{ 13\times 13} }{\overline{2\times 2}\times\overline{ 5\times 5}}}$

= $\frac{2\times 13}{2\times 5}$

= $\frac{26}{10}=2.6$

8) Fin the square root of $1\frac{4}{5}$ correct upto 3 decimal .
Solution : In order to approximate the square root of $1\frac{4}{5}$ upto 3 decimal places , we have to calculate the square root upto 4 decimal places.
$\sqrt{1\frac{4}{5}}=\sqrt{1.8}$

$\therefore$ The square root of $1\frac{4}{5}$ is approximated to 1.342 .

9) What is the smallest number that must be subtracted from 6098 in order in order to get a perfect square ?
Solution: Square root of 6098

As we are left with 14 as remainder, if 14 is subtracted from 6098, we would get 6098-14 =6084 , a perfect square as $78^{2} = 6084$ .

10) What is the smallest number that must be added to 7190 to get a perfect square ?
Solution : Square root of 7190

The remainder indicates that $84^{2}< 7190 < 85^{2}$ . As we have to add to 7190 in order to get a perfect square we have to add the difference of $85^{2}$ and 7190 or 7225-7190 = 35
Thus if 35 is added to 7190 , we get a perfect square 7225 or $85^{2}$ .

Square and square roots class 8 extra questions

1) Fill in the blanks :
(i) The square of 0.5 is ………….
(ii) The square root of 0.49 is ………
(iii) Square of an odd number is always …………
(iv) Square of an even number is always …………
(v)Square of every integer is …………
(vi)Square of every negative integer is …………..
(vii) A number whose exact square root can be obtained, is called a …………….
(viii) The square of a proper fraction is ………………… the proper fraction.
(ix) The square of a improper fraction is ………………… the improper fraction.
(x) There are ……….. non perfect square numbers between the squares of the numbers n and (n + 1).

2) Find the value of the following:
(i) $\sqrt{\frac{9}{16}}$ (ii) $\sqrt{1\frac{21}{100}}$ (iii) $\sqrt{6.25}$

3) Find the square root of 7056 .

4) Find the square root of $3\frac{109}{225}$

5) Find the square root of $1\frac{4}{5}$ correct upto 3 decimal .

6) Find the square of $1\frac{3}{5}$ .

7)Find the square root of 6.76 .

8) Find the square root of 101124 by division method .

9) Find the square root of 464.8336 by division method .

10) Find the greatest and smallest and greatest 6 -digit numbers that are perfect square .

11) Find the smallest number by which 980 should be multiplied to make it a perfect square .

12) Find the smallest number by which 12168 should be divided to make it a perfect square .

13) Which of the following is a perfect square (i) $\sqrt{25}$ (ii) $\sqrt{\frac{4}{9}}$ (iii) $\sqrt{0.16}$ (iv) $\sqrt{5}$ (v) $\sqrt{3}$

14) Find the square root of 5 correct to 2 decimal places. Hence find the value of $\sqrt{\frac{3+\sqrt{5}}{3-\sqrt{5}}}$ correct to three significant digits.