NCERT Solutions for Class 8 Maths Chapter 6- Squares and Square Roots Exercise 6.4
NCERT Solutions for Class 8 Maths Chapter 6- Squares and Square Roots Exercise 6.4 are comprehensive and exhaustive solutions. They have been meticulously prepared by Vedantu’s subject matter experts who have vast experience in the education industry. The Class 8 Maths Exercise 6.4 Solution has been prepared according to the guidelines of CBSE. NCERT Solution for Class 8 Maths Chapter 6 Exercise 6.4 can be conveniently downloaded in free PDF formats. that will allow you to study when you are offline as well.
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Access NCERT Solutions for Mathematics Chapter 6 – Squares and Square Roots
Exercise 6.4
1. Find the square root of each of the following numbers by division method.
2304
Ans: The square root of the given number is
$\quad\quad 48 \quad$ | |
$\quad 4\quad$ | $\overline{23}\:\overline{04}\\$ $ -16\qquad $ |
$\quad 88\quad$ | ${\;}704\\ $ $ {\;}704 $ |
$\qquad\quad 0\quad$ |
$\Rightarrow \text{ }\sqrt{2304} = 48$
4489
Ans: The square root of the given number is
$\quad\quad 67 \quad$ | |
$\quad 6\quad$ | $ \overline{44}\:\overline{89}\\ $ $ -36\qquad $ |
$\quad 127\quad$ | ${\;} 889\\ $ $ {\;} 889 $ |
$\qquad\quad 0\quad$ |
$\Rightarrow \text{ }\sqrt{4489} = 67$.
3481
Ans: The square root of the given number is
$\quad\quad 59 \quad$ | |
$\quad 5\quad$ | $\overline{34}\:\overline{81}\\ $ $ -25\qquad $ |
$\quad 109\quad$ | ${\;} 981\\ $ $ {\;} 981 $ |
$\qquad\quad 0\quad$ |
$\Rightarrow \text{ }\sqrt{3481} = 59$
529
Ans: The square root of the given number is
$\quad \quad 23 \quad$ | |
$\quad 2\quad$ | $\overline{5}\:\overline{29}\\ $ $ -4\qquad $ |
$\quad 43\quad$ | $ {} 129\\ $ $ {} 129 $ |
$\qquad\quad 0\quad$ |
$\Rightarrow \text{ }\sqrt{529} = 23$.
3249
Ans: The square root of the given number is
$\quad\quad 57 \quad$ | |
$\quad 5\quad$ | $ \overline{32}\:\overline{49}\\ $ $ -25\qquad $ |
$\quad 107\quad$ | ${\;} 749\\$ $ {\;} 749 $ |
$\qquad\quad 0\quad$ |
$\Rightarrow \text{ }\sqrt{3249} = 57$
1369
Ans: The square root of the given number is
$\quad\quad 37 \quad$ | |
$\quad 3\quad$ | $ \overline{13}\:\overline{69}\\ $ $ -9\qquad $ |
$\quad 67\quad$ | $ {\;} 469\\ $ $ {\;} 469 $ |
$\qquad\quad 0\quad$ |
$\Rightarrow \text{ }\sqrt{1369} = 37$
5776
Ans: The square root of the given number is
$\quad\quad 76 \quad$ | |
$\quad 7\quad$ | $ \overline{57}\:\overline{76}\\ $ $ -49\qquad $ |
$\quad 146\quad$ | ${\;} 876\\ $ $ {\;} 876 $ |
$\qquad\quad 0\quad$ |
$\Rightarrow \text{ }\sqrt{5776} = 76$.
7921
Ans: The square root of the given number is
$\quad\quad 30 \quad$ | |
$\quad 8\quad$ | $ \overline{79}\:\overline{21}\\ $ $ -64\qquad $ |
$\quad 169\quad$ | $ {\;} 1521\\ $ $ {\;} 1521 $ |
$\qquad\quad 0\quad$ |
$\Rightarrow \text{ }\sqrt{7921} = 89$
576
Ans: The square root of the given number is
$\quad\quad 24 \quad$ | |
$\quad 2\quad$ | $\overline{5}\:\overline{76}\\ $ $ -4\qquad $ |
$\quad 44\quad$ | $ {\;} 176\\ $ $ {\;} 176 $ |
$\qquad\quad 0\quad$ |
$\Rightarrow \text{ }\sqrt{576} = 24$
1024
Ans: The square root of the given number is
$\quad\quad 32 \quad$ | |
$\quad 3\quad$ | $\overline{10}\:\overline{24}\\$ $ -9\qquad $ |
$\quad 62\quad$ | $ {\;} 124\\ $ $ {\;} 124 $ |
$\qquad\quad 0\quad$ |
$\Rightarrow \text{ }\sqrt{1024} = 32$
3136
Ans:The square root of the given number is
$\quad\quad 56 \quad$ | |
$\quad 5\quad$ | $ \overline{31}\:\overline{36}\\ $ $ -25\qquad $ |
$\quad 106\quad$ | $ {\;} 636\\ $ $ {\;} 636 $ |
$\qquad\quad 0\quad$ |
$\Rightarrow \text{ }\sqrt{3136} = 56$
900
Ans: The square root of the given number is
$\quad \quad 30 \quad$ | |
$\quad 3\quad$ | $\overline{9}\:\overline{00}\\ $ $-9\qquad $ |
$\quad 60\quad$ | $ {\;} 00\\ $ $ {\;} 00 $ |
$\qquad\quad 0\quad$ |
$\Rightarrow \text{ }\sqrt{900} = 30$
2. Find the number of digits in the square root of each of the following numbers (without any calculation).
64
Ans: Starting from the right side, take the numbers into pairs and place the bar above the number, then we get
$64 = \overline{64}$
There is only one bar (one pair) in the given number, so the number of digits in the square root will be one digit.
144
Ans: Starting from the right side, take the numbers into pairs and place the bar above the number, then we get
\[144 = \overline{1}\overline{44}\]
There are only two bar (two pair) in the given number, so the number of digits in the square root will be two-digit.
4489
Ans: Starting from the right side, take the numbers into pairs and place the bar above the number, then we get
$4489 = \overline{44}\overline{89}$
There is only two bar (two pair) in the given number, so the number of digits in the square root will be two-digit.
27225
Ans: Starting from the right side, take the numbers into pairs and place the bar above the number, then we get
$27225 = \overline{2}\overline{72}\overline{25}$
There is only three bar (three pair) in the given number, so the number of digits in the square root will be three-digit.
390625
Ans: Starting from the right side, take the numbers into pairs and place the bar above the number, then we get
$390625 = \overline{39}\overline{06}\overline{25}$
There is only three bar (three pair) in the given number, so the number of digits in the square root will be three-digit.
3. Find the square root of the following decimal numbers.
\[~\mathbf{2}.\mathbf{56}\]
Ans: The square root of the given number is
$\quad\quad 1.6 \quad$ | |
$\quad 1\quad$ | $\overline{2.}\:\overline{56}\\ $ $ -1\qquad $ |
$\quad 26\quad$ | ${\;} 156\\ $ $ {\;} 156 $ |
$\qquad\quad 0\quad$ |
$\Rightarrow \text{ }\sqrt{2.56} = 1.6$.
\[\mathbf{7}.\mathbf{29}\]
Ans: The square root of the given number is
$\quad\quad 2.7 \quad$ | |
$\quad 2\quad$ | $ \overline{7.}\:\overline{29}\\ $ $ -4\qquad $ |
$\quad 47\quad$ | ${\;} 329\\ $ $ {\;} 329 $ |
$\qquad\quad 0\quad$ |
$\Rightarrow \text{ }\sqrt{7.29} = 2.7$.
51.84
Ans: The square root of the given number is
$\quad\quad 7.2 \quad$ | |
$\quad 7\quad$ | $\overline{51.}\:\overline{84}\\ $ $ -49\qquad $ |
$\quad 142 \quad$ | ${\;} 284\\ $ $ {\;} 284 $ |
$\qquad\quad 0\quad$ |
$\Rightarrow \text{ }\sqrt{51.84} = 7.2$.
42.25
Ans: The square root of the given number is
$\quad\quad 6.5 \quad$ | |
$\quad 6\quad$ | $\overline{42}\:\overline{.25}\\ $ $ -36\qquad $ |
$\quad 125\quad$ | $ {\;} 625\\ $ $ {\;} 625 $ |
$\qquad\quad 0\quad$ |
$\Rightarrow \text{ }\sqrt{42.25} = 6.5$.
31.36
Ans: The square root of the given number is
$\quad\quad 5.6 \quad$ | |
$\quad 5\quad$ | $ \overline{31}\:\overline{.36}\\ $ $ -9\qquad $ |
$\quad 106\quad$ | $ {\;} 636\\$ $ {\;} 636 $ |
$\qquad\quad 0\quad$ |
$\Rightarrow \text{ }\sqrt{31.36} = 5.6$.
4. 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.
$402$
Ans: The square root of the given number is
$\quad\quad 20 \quad$ | |
$\quad 2\quad$ | $\overline{4}\:\overline{02}\\$ $-4\qquad $ |
$\quad 40\quad$ | ${\;} 002\\$ $ {\;} 002 $ |
$\qquad\quad 0\quad$ |
The remainder of the long division method is$2$.
It shows that the square of $20$ is less than the square root of $402$ .
Therefore a perfect square will be get by subtracting $2$ from the$402$.
$\therefore \Rightarrow $The required perfect square $= 402 - 2 = 400$
The square root of the perfect square$\sqrt{400} = 20$.
1989
Ans: The square root of the given number is
$\quad\quad 44 \quad$ | |
$\quad 4\quad$ | $\overline{19}\:\overline{89}\\ $ $ -16\qquad $ |
$\quad 84\quad$ | $ {\;} 389\\ $ $ {\;} 336 $ |
$\qquad\quad 53\quad$ |
The remainder of the long division method is $53$.
It shows that the square of $44$ is less than the square root of $1989$ .
Therefore a perfect square will be obtained by subtracting $53$ from the$1989$.
$\therefore \Rightarrow $The required perfect square $= 1989 - 53 = 1936$
The square root of the perfect square$\sqrt{1989} = 44$.
3250
Ans: The square root of the given number is
$\quad\quad 57 \quad$ | |
$\quad 5\quad$ | $ \overline{32}\:\overline{50}\\ $ $ -25\qquad $ |
$\quad 107\quad$ | $ {\;} 750\\ $ $ {\;} 749 $ |
$\qquad\quad 1\quad$ |
The remainder of the long division method is $1$.
It shows that the square of $57$ is less than the square root of $3250$ .
Therefore a perfect square will be get by subtracting $2$ from the $3250$.
$\therefore \Rightarrow $The required perfect square $= 3250 - 1 = 3249$
The square root of the perfect square $\sqrt{3249} = 57$.
825
Ans: The square root of the given number is
$\quad\quad 28 \quad$ | |
$\quad 2\quad$ | $\overline{8}\:\overline{25}\\ $ $ -4\qquad $ |
$\quad 48\quad$ | $ {\;} 425\\ $ $ {\;} 384 $ |
$\qquad\quad 41\quad$ |
The remainder of the long division method is $41$.
It shows that the square of $28$ is less than the square root of $825$ .
Therefore a perfect square will be get by subtracting $41$ from the $825$.
$\therefore \Rightarrow $The required perfect square $= 825 - 41 = 784$
The square root of the perfect square $\sqrt{784} = 28$.
4000
Ans: The square root of the given number is
$\quad\quad 63 \quad$ | |
$\quad 6\quad$ | $ \overline{40}\:\overline{00}\\ $ $ -36\qquad $ |
$\quad 123\quad$ | $ {\;} 400\\ $ $ {\;} 369 $ |
$\qquad\quad 31\quad$ |
The remainder of the long division method is $31$.
It shows that the square of $63$ is less than the square root of $4000$ .
Therefore a perfect square will be get by subtracting $31$ from the $4000$.
$\therefore \Rightarrow $The required perfect square $= 4000 - 31 = 3969$
The square root of the perfect square $\sqrt{3969} = 63$.
5. Find the least number which must be added to each of the following numbers so as to get a perfect square. Also, find the square root of the perfect square so obtained.
525
Ans: The square root of the given number is
$\quad\quad 22 \quad$ | |
$\quad 2\quad$ | $ \overline{5}\:\overline{25}\\ $ $ -4\qquad $ |
$\quad 42\quad$ | $ {\;} 125\\ $ $ {\;} 84 $ |
$\qquad\quad 41\quad$ |
The remainder of the long division method is $41$.
It shows that the square of $22$ is less than the square root of $525$ .
${{22}^{2}} = 484 \And {{23}^{2}} = 529$
We need to find the least number, which must be added to the given number to get a perfect square.
Therefore a perfect square will be get by subtracting ${{23}^{2}}$ from the $525$.
$\therefore \Rightarrow $The required perfect square $= 529 - 525 = 4$
The square root of the perfect square $\sqrt{529} = 23$.
1750
Ans: The square root of the given number is
$\quad\quad 41 \quad$ | |
$\quad 4\quad$ | $ \overline{17}\:\overline{50}\\ $ $ -16\qquad $ |
$\quad 51\quad$ | $ {\;} 150\\ $ $ {\;} 81 $ |
$\qquad\quad 69\quad$ |
The remainder of the long division method is \[69\].
It shows that the square of $41$ is less than the square root of $1750$ .
${{41}^{2}} = 1681 \And {{42}^{2}} = 1764$
We need to find the least number, which must be added to the given number so as to get a perfect square.
Therefore a perfect square will be get by subtracting ${{42}^{2}}$ from the $1750$.
$\therefore \Rightarrow $The required perfect square $= 1764 - 1750 = 14$
The square root of the perfect square $\sqrt{1764} = 42$.
252
Ans: The square root of the given number is
$\quad\quad 15 \quad$ | |
$\quad 1\quad$ | $ \overline{2}\:\overline{52}\\ $ $ -1\qquad $ |
$\quad 25\quad$ | $ {\;} 152\\ $ $ {\;} 125 $ |
$\qquad\quad 27\quad$ |
The remainder of the long division method is \[27\].
It shows that the square of $15$ is less than the square root of $252$ .
${{15}^{2}} = 225 \And {{16}^{2}} = 256$
We need to find the least number, which must be added to the given number so as to get a perfect square.
Therefore a perfect square will be get by subtracting ${{16}^{2}}$ from the $252$.
$\therefore \Rightarrow $The required perfect square $= 256 - 252 = 4$
The square root of the perfect square $\sqrt{1256} = 16$.
1825
Ans: The square root of the given number is
$\quad\quad 42 \quad$ | |
$\quad 4\quad$ | $ \overline{18}\:\overline{25}\\ $ $ -16\qquad $ |
$\quad 82\quad$ | $ {\;} 225\\ $ $ {\;} 164 $ |
$\qquad\quad 61\quad$ |
The remainder of the long division method is \[61\].
It shows that the square of $42$ is less than the square root of $1825$ .
${{42}^{2}} = 1764 \And {{43}^{2}} = 1849$
We need to find the least number, which must be added to the given number so as to get a perfect square.
Therefore a perfect square will be get by subtracting ${{43}^{2}}$ from the $1825$.
$\therefore \Rightarrow $The required perfect square $= 1849 - 1825 = 24$
The square root of the perfect square $\sqrt{1849} = 43$.
6412
Ans: The square root of the given number is
$\quad\quad 80 \quad$ | |
$\quad 8\quad$ | $ \overline{64}\:\overline{12}\\ $ $ -64\qquad $ |
$\quad 160\quad$ | $ {\;} 012\\$ $ {\;} 0 $ |
$\qquad\quad 12\quad$ |
The remainder of the long division method is \[12\].
It shows that the square of $15$ is less than the square root of $6412$ .
${{80}^{2}} = 6400 \And {{81}^{2}} = 6561$
We need to find the least number, which must be added to the given number so as to get a perfect square.
Therefore a perfect square will be get by subtracting ${{81}^{2}}$ from the $6412$.
$\therefore \Rightarrow $The required perfect square $= 6561 - 6412 = 149$
The square root of the perfect square $\sqrt{6561} = 81$.
6. Find the length of the side of a square whose area is $441 {{m}^{2}}$.
Ans: Let us take the length of the side of the square to be $x m$.
Area of Square $= {{\left( x \right)}^{2}} = 441 {{m}^{2}}$.
$x = \sqrt{441}$
The square root of $x $is
$\quad\quad 21 \quad$ | |
$\quad 2\quad$ | $ \overline{4}\:\overline{41}\\ $ $ -4\qquad $ |
$\quad 41\quad$ | $ {\;} 041\\ $ $ {\;} 041 $ |
$\qquad\quad 0\quad$ |
$\therefore \Rightarrow x = 21 m$.
So, The length of the side of a square is $21 m$.
7. In a right triangle
$ ABC, \therefore B\text{ }=\text{ }90{}^\circ . \\ $
If \[AB = 6 cm , BC = 8 cm, \]find \[AC\]
Ans:
\[\Delta ABC\]is a right angle at \[B.\]
By Pythagoras theorem,
\[A{{C}^{2}} = A{{B}^{2}} + B{{C}^{2}}\]
\[A{{C}^{2}} = {{\left( 6 cm \right)}^{2}} + {{\left( 8 cm \right)}^{2}}\]
\[A{{C}^{2}} = \left( 36 + 64 \right) c{{m}^{2}}\]
\[A{{C}^{2}} = \left( 100 \right)c{{m}^{2}}\]
\[AC = \left( \sqrt{100} \right)cm = \left( \sqrt{10 \times 10} \right)cm\]
\[AC = 10cm\].
If \[AC = 13 cm, BC = 5 cm,\] find \[AB\]
Ans:
\[\Delta ABC\]is a right angle at \[B.\]
By Pythagoras theorem,
\[A{{C}^{2}} = A{{B}^{2}} + B{{C}^{2}}\]
\[{{\left( 13 \right)}^{2}} = A{{B}^{2}} + {{\left( 5 cm \right)}^{2}}\]
\[A{{B}^{2}} = {{\left( 13 cm \right)}^{2}} - {{\left( 5 cm \right)}^{2}}\]
\[A{{B}^{2}} = \left( 169 - 25 \right)c{{m}^{2}}\]
\[A{{B}^{2}} = \left( 144 \right)c{{m}^{2}}\]
\[AC = \left( \sqrt{144} \right)cm = \left( \sqrt{12 \times 12} \right)cm\]
\[AB = 12 cm\]
8. A gardener has $1000$plants. He wants to plant these in such a way that the number of rows and the number of columns remain the same. Find the minimum number of plants he needs more for this.
Ans:
The gardener has $1000$ plants and it has the same number of rows and the same number of columns.
The square root of 1000 is
$\quad\quad 31 \quad$ | |
$\quad 3\quad$ | $ \overline{10}\:\overline{00}\\ $ $ -9\qquad $ |
$\quad 61\quad$ | $ {\;} 100\\ $ $ {\;} 061 $ |
$\qquad\quad 39\quad$ |
The remainder of the long division method is \[39\].
It shows that the square of $31$ is less than the square root of $1000$ .
${{31}^{2}} = 961 \And {{32}^{2}} = 1024$
We need to find the least number, which must be added to the given number so as to get a perfect square.
Therefore a perfect square will be get by subtracting ${{32}^{2}}$ from the$1000$.
$\therefore \Rightarrow $The required perfect square $= 1024 - 1000 = 24$
Hence, The required number of plants is $24$.
9. These are $500$ children in a school. For a P.T. drill, they have to stand in such a manner that the number of rows is equal to the number of columns. How many children would be left out in this arrangement?
Ans:
It is given that there are $500$ children in a school.
For a P.T. drill, they have to stand in such a manner that the number of rows is equal to the number of columns.
To find that the number of children left in this arrangement,
$\quad\quad 22 \quad$ | |
$\quad 2\quad$ | $ \overline{5}\:\overline{00}\\ $ $ -4\qquad $ |
$\quad 42\quad$ | $ {\;} 100\\ $ $ {\;} 084 $ |
$\qquad\quad 16\quad$ |
The remainder of the long division method is$16$.
It shows that the square of $22$ is less than the square root of $500$ .
Therefore a perfect square will be get by subtracting $16$ from the $500$.
$\therefore \Rightarrow $ The required perfect square $= 500 - 16 = 484$
The square root of the perfect square $16$.
Then the number of children would be left out in this arrangement is $16$
Maths NCERT Solutions Class 8 Chapter 6 Exercise 6.4
Class 8 is one of the most conceptually vital formative years for a student. It is important to have a good grip on the topics and subtopics learned in class 8 to understand more complex concepts in subsequent years. Concepts that you learn in Class 8 Maths will help you build a strong conceptual foundation. This is the right time to focus on learning Maths fundamentals the most. Class 8 is pretty much the building block of your future. Class 8 Maths Chapter 6- Squares and Square Roots is a very important chapter from the exam point of view. Prior knowledge of exponents shall help you learn this chapter well.
Chapter 6 is quite lengthy, and it is not easy to learn this chapter all on your own. You are bound to face challenges while solving NCERT Practice problems on your own. Hence it is advisable that you learn to solve all the solved examples, practice exercises, and relevant questions with the help of NCERT Solution of Class 8 Maths Chapter 6. This way you will rid yourself of all the ambiguity related to squares and square roots.
Class 8 Maths Chapter 6 deals with a number of important concepts such as Square Number, Square Roots, Properties of Square Numbers, Numbers between Square Numbers, Adding Odd Numbers, Sum of Consecutive Natural Numbers, Product of Two Consecutive Odd or Even Numbers, Finding the Square of A Number, Pythagorean Triplets, Finding Square Root Through Repeated Subtraction, Finding Square Root Through Prime Factorization, Finding Square Root Through Division Method, Square Roots of Decimals, and Estimating Square Roots
Class 8 Maths Chapter 6 Exercise 6.4 deals with Finding the Square Root Through Division Method, Finding Square Roots of Decimals, and Estimation of Square Roots. NCERT exercises mostly follow the same pattern as the CBSE question papers. Exercise 6.4 Class 8 also follows the same pattern. It includes all possible question types from this section that can be asked in your CBSE Class 8 Maths exams. They also include questions that cover all the topics in this section to facilitate splendid conceptual understanding.
Advantages of CBSE Maths NCERT Solutions Class 8 Chapter 6 Exercise 6.4
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