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Important Questions for CBSE Class 8 Maths Chapter 6 - Squares and Square Roots

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CBSE Class 8 Maths Important Questions for Squares and Square Roots - Free PDF Download

Important Questions for CBSE Class 8 Chapter 6 Square and Square Roots with solutions are available on Vedantu’s online platform. Our experienced teachers from the mathematics department have developed the important questions with answers for the chapter as per the latest syllabus of CBSE Board. These important questions with solutions will definitely benefit all students with revision and mastering the topic. Students can also reach out to Vedantu to get Important Questions for Class 8 Maths for all other chapters. Vedantu is a platform that provides free (CBSE) NCERT Solution and other study materials for students. You can also download NCERT Solutions for Class 8 Maths and NCERT Solution for Class 8 Science to help you to revise the complete syllabus and score more marks in your examinations.


Topics Covered in Class 8 Maths Chapter 6 Squares and Square Roots 

To solve the important questions, students must have a thorough knowledge of the concepts discussed in this chapter. So, before jumping to the important questions, let us discuss here some of the important topics and subtopics covered in this chapter. 

  1. Introduction to Square Numbers

  2. Properties of Square Numbers

  3.  Some More Interesting Patterns

  1. Adding triangular numbers

  2. Numbers between square numbers

  3. Adding odd numbers

  4. A sum of consecutive natural numbers

  5. Product of two consecutive even or odd natural numbers

  6. Some more patterns in square numbers

  1. Finding the Square of a Number

  1. Other patterns in squares

  2. Pythagorean triplets

  1. Square Roots

  1. Finding square roots

  2. Finding square root through repeated subtraction

  3. Finding square root through prime factorisation

  4. Finding square root by division method

  1. Square Roots of Decimals 

  2. Estimating Square Root

Chapter 6 Square and Square Roots Important Questions PDF

Very Short Answer Questions (1 Mark)

1. A number ending in __, __, __ or __ is never a perfect square.

Ans: \[2,3,7,8\]


2. If a number divided by 3 leaves a remainder 2, it is not a perfect square. Say true or false.

Ans: True


3. \[{{\mathbf{8}}^{\mathbf{2}}}\]

Ans: ${{8}^{2}}=8\times 8=64$


4. The square of even number is always ___.

Ans: Even


5. The square of odd number is always ___.

Ans: Odd


6. If a number when divided by 4 leaves a remainder 2 or 3, then it is a perfect square. Say true or false.

Ans: False


7. The sum of first $n$ odd natural numbers is ___.

Ans: ${{n}^{2}}$


8. The difference of two perfect squares is a perfect square. Say true or false.

Ans: False


9. The product of two perfect square is a perfect square. Say true or false.

Ans: True


Short Answer Questions (2 Marks)

10. Is \[\mathbf{496}\] a perfect square?

Ans: 

$\begin{align}& 2\left| \!{\underline {\,496 \,}} \right.  \\ & 2\left| \!{\underline {\,148 \,}} \right.  \\  & 2\left| \!{\underline {\,124 \,}} \right.  \\ & 2\left| \!{\underline {\,62 \,}} \right.  \\  & \text{   }31 \\  & \Rightarrow 496=2\times 2\times 2\times 2\times 31 \\ \end{align}$

Hence, the number cannot be expressed as pairs of factors.


11.\[~\mathbf{4000}\] is never a perfect square. Why? Ans: The number $4000$ has three zeros. Hence, it can never be a perfect square.


12. \[\mathbf{{{\left( n+1 \right)}^{2}}-{{n}^{2}}=}\]?

Ans: 

\[\begin{align}  & {{\left( n+1 \right)}^{2}}-{{n}^{2}} \\  & =\left( n+1+n \right)\left( n+1-n \right) \\  & =\left\{ \left( n+1 \right)+n \right\} \\ \end{align}\]


13. Write the general form of Pythagorean triplet using m, n, p.

Ans: For every natural number $m>1$, we have $\left( 2m,{{m}^{2}}-1,{{m}^{2}}+1 \right)$ as Pythagorean triplet.


14. State whether the square of given number is even or odd.

  1. \[\mathbf{524}\]

  2. \[\mathbf{655}\]

Ans:

  1. The square of $524$ is even.

  2. The square of $655$ is odd.


15. Write a Pythagorean triplet whose smallest number is \[\mathbf{9}\].

Ans: 

$\begin{align}& {{a}^{2}}=b+c \\ & c-b=1 \\ & \Rightarrow {{9}^{2}}=81=40+41 \\ \end{align}$

So, the Pythagorean triplet is $\left( 9,40,41 \right)$.


Short Answer Questions (3 Marks)

16. Is \[\mathbf{5292}\] a perfect square?

Ans:

$\begin{align}& 2\left| \!{\underline {\,5292 \,}} \right.  \\ & 2\left| \!{\underline {\,2646 \,}} \right.  \\ & 2\left| \!{\underline {\,1323 \,}} \right.  \\  & 3\left| \!{\underline {\, 441 \,}} \right.  \\ & 3\left| \!{\underline {\,147 \,}} \right.  \\ & 7\left| \!{\underline {\,49 \,}} \right.  \\ & \text{   }7 \\ & \Rightarrow 5292=2\times 2\times 2\times 3\times 3\times 7\times 7 \\ \end{align}$

The number cannot be expressed as pairs of factors.

Hence, it is not a perfect square.


17. Solve \[~{{\left( \mathbf{36} \right)}^{\mathbf{2}}}-{{\left( \mathbf{35} \right)}^{\mathbf{2}}}\].

Ans:

${{36}^{2}}-{{35}^{2}}=36+35=71$


18. Without adding, find the sum of \[\mathbf{1}+\mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{9}+\mathbf{11}+\mathbf{13}+\mathbf{15}+\mathbf{17}+\mathbf{19}+\mathbf{21}\].

Ans:

$\text{Sum of first }11\text{ odd numbers}={{11}^{2}}=121$


19. \[{{\left( \mathbf{509} \right)}^{\mathbf{2}}}=\]?

Ans:

$\begin{align}& {{509}^{2}} \\ & ={{\left( 500+9 \right)}^{2}} \\ & ={{500}^{2}}+{{9}^{2}}+2\times 500\times 9 \\ & =250000+81+9000 \\ & =259081 \\ \end{align}$


20. Evaluate ${{\left( \mathbf{691} \right)}^{\mathbf{2}}}$.

Ans:

 $\begin{align}& {{691}^{2}} \\ & ={{\left( 700-9 \right)}^{2}} \\ & ={{700}^{2}}+{{9}^{2}}-2\times 700\times 9 \\  & =490000+81-12600 \\  & =477481 \\ \end{align}$


21. Evaluate \[\mathbf{39}\times \mathbf{41}\].

Ans:

 $\begin{align}& 39\times 41 \\ & =\left( 40-1 \right)\left( 40+1 \right) \\ & ={{40}^{2}}-{{1}^{2}} \\ & =1600-1 \\ & =1599 \\ \end{align}$


22. Express \[\mathbf{121}\] as sum of \[\mathbf{11}\] odd numbers.

Ans: We know that,

$\begin{align}& \text{Sum of first }n\text{ odd numbers}={{n}^{2}} \\ & \text{Here, }{{11}^{2}}=121 \\ \end{align}$

Hence, expressing $121$ as sum of $11$ odd numbers,

$1+3+5+7+9+11+13+15+17+19+21$


Long Answer Questions (4-5 Marks)

23. By what least number should 720 be multiplied to get a perfect square number? Also, find the number whose square is the new number.

Ans:

 $\begin{align}& 2\left| \!{\underline {\,720 \,}} \right.  \\ & 2\left| \!{\underline {\,360 \,}} \right.  \\ & 2\left| \!{\underline {\,180 \,}} \right.  \\ & 2\left| \!{\underline {\,90 \,}} \right.  \\ & 3\left| \!{\underline {\,45 \,}} \right.  \\ & 3\left| \!{\underline {\,15 \,}} \right.  \\ & \text{    }3 \\  & \Rightarrow 720=2\times 2\times 2\times 2\times 3\times 3\times 5 \\ \end{align}$

Here, $5$ doesn’t have a pair. If the number is multiplied by $5$, $720$ becomes a perfect square.

Hence, the new number formed after multiplying is $720\times 5=3600$.


24. By what least number should \[\mathbf{2527}\] be divided to get a perfect square number? Find the number whose square is the new number.

Ans: 

\[\begin{align}& 7\left| \!{\underline {\,2527 \,}} \right.  \\ & 19\left| \!{\underline {\,361 \,}} \right.  \\  & \text{      }19 \\  & \Rightarrow 2527=7\times 19\times 19 \\ \end{align}\]

Here, $7$ doesn’t have a pair. If the number is divided by $7$, $2527$ becomes a perfect square.

Hence, the new number formed after multiplying is $2527\div 7=361$.


25. Find the largest number of \[\mathbf{4}\] digits which is a perfect square.

Ans: The largest $4$digit number is $9999$.

${{99}^{2}}$ is less than $9999$ by $198$.

Hence, the required number is $9999-198=9801$.


26. Find the sum of \[\mathbf{49}\] by column method.

Ans: Given number$=49$

$a=4,b=9$

$\begin{array}{ccc} \text { I } & \text { II } & \text { III } \\ a^{2} & 2 \times a \times b & b^{2} \\ 4^{2} & 2 \times 4 \times 9 & 9^{2} \\ 16 & 72 & 81 \\+8 & +8 & \\ \hline 24 & {80} \end{array}$

Hence, ${{49}^{2}}=2401$.


27. Find the squares of \[\mathbf{41}\] using diagonal method.

Ans:


Squares of \[\mathbf{41}\] using diagonal method

 

Hence, ${{41}^{2}}=1681$


28. Find the square root of \[\mathbf{441}\].

Ans

$\begin{align}& 3\left| \!{\underline {\,441 \,}} \right.  \\ & 3\left| \!{\underline {\,147 \,}} \right.  \\ & 7\left| \!{\underline {\,49 \,}} \right.  \\  & \text{    }7 \\ & \Rightarrow \sqrt{441}=\sqrt{3\times 3\times 7\times 7} \\ & =3\times 7 \\  & =21 \\ \end{align}$


29. Find the square root of \[\mathbf{11025}\].

Ans: 

$\begin{align}& 3\left| \!{\underline {\,11025 \,}} \right.  \\ & 3\left| \!{\underline {\,3675 \,}} \right.  \\ & 5\left| \!{\underline {\,1225 \,}} \right.  \\ & 5\left| \!{\underline {\,245 \,}} \right.  \\ & 7\left| \!{\underline {\,49 \,}} \right.  \\ & \text{    }7 \\ & \Rightarrow \sqrt{11025}=\sqrt{3\times 3\times 5\times 5\times 7\times 7} \\ & =3\times 5\times 7 \\  & =105 \\ \end{align}$


30. In a movie theatre, the number of rows is equal to the number of chairs in each row. If the capacity of the theatre is 2025. Find the number of chairs in each row.

Ans

$\begin{align}& 5\left| \!{\underline {\,2025 \,}} \right.  \\ & 5\left| \!{\underline {\,405 \,}} \right.  \\ & 3\left| \!{\underline {\,81 \,}} \right.  \\ & 3\left| \!{\underline {\,27 \,}} \right.  \\ & 3\left| \!{\underline {\,9 \,}} \right.  \\ & 3\left| \!{\underline {\,3 \,}} \right.  \\ & \text{  1} \\ & \Rightarrow \sqrt{2025}=\sqrt{5\times 5\times 3\times 3\times 3\times 3} \\ & =5\times 3\times 3 \\ & =45 \\ \end{align}$


31. Find the square root of \[\mathbf{55696}\] by long division method.

Ans: 


The square root of \[\mathbf{55696}\] by long division method


Hence, $\sqrt{55696}=236$


32. \[\sqrt{\mathbf{12}.\mathbf{25}}\] and $\sqrt{\frac{\mathbf{625}}{\mathbf{729}}}$.

Ans:

$\sqrt{12.25}$ can be written as,

$\sqrt{12.25}=\sqrt{\frac{1225}{100}}$

For numerator,

$\begin{align}& 5\left| \!{\underline {\,1225 \,}} \right.  \\ & 5\left| \!{\underline {\,245 \,}} \right.  \\ & 7\left| \!{\underline {\,49 \,}} \right.  \\ & \text{     7} \\ & \Rightarrow \sqrt{1225}=\sqrt{5\times 5\times 7\times 7} \\ & =5\times 7 \\  & =35 \\ \end{align}$

For denominator,

$\begin{align}& 2\left| \!{\underline {\,100 \,}} \right.  \\ & 2\left| \!{\underline {\,50 \,}} \right.  \\ & 5\left| \!{\underline {\,25 \,}} \right.  \\ & 5\left| \!{\underline {\, 5 \,}} \right.  \\ & \text{  1} \\ & \Rightarrow \sqrt{100}=\sqrt{2\times 2\times 5\times 5} \\ & =2\times 5 \\ & =10 \\ \end{align}$

Hence, 

$\begin{align}& \sqrt{12.25}=\sqrt{\frac{1225}{100}} \\ & =\frac{\sqrt{1225}}{\sqrt{100}} \\ & =\frac{35}{10} \\ & =3.5 \\ \end{align}$

Square root of $\sqrt{\frac{625}{729}}$,

For numerator,

$\begin{align}& 5\left| \!{\underline {\,625 \,}} \right.  \\ & 5\left| \!{\underline {\,125 \,}} \right.  \\ & 5\left| \!{\underline {\,25 \,}} \right.  \\ & \text{     5} \\ & \Rightarrow \sqrt{625}=\sqrt{5\times 5\times 5\times 5} \\ & =5\times 5 \\ & =25 \\ \end{align}$

For denominator,

$\begin{align}& 3\left| \!{\underline {\,729 \,}} \right.  \\ & 3\left| \!{\underline {\,243 \,}} \right.  \\ & 3\left| \!{\underline {\,81 \,}} \right.  \\ & 3\left| \!{\underline {\,27 \,}} \right.  \\ & 3\left| \!{\underline {\,9 \,}} \right.  \\ & 3\left| \!{\underline {\,3 \,}} \right.  \\ & \text{  1} \\ & \Rightarrow \sqrt{729}=\sqrt{3\times 3\times 3\times 3\times 3\times 3} \\ & =3\times 3\times 3 \\ & =27 \\ \end{align}$

Hence, 

\[\begin{align}& \sqrt{\frac{625}{729}}=\frac{\sqrt{625}}{\sqrt{729}} \\ & =\frac{25}{27} \\ \end{align}\]

Chapter 6 Square and Square Roots Important Questions PDF

In this chapter, you will learn the concepts of square and square roots. There are many important questions given for the chapter in the PDF version free that will give you an idea of what kind of question you can expect in exams. You can use the pdf of the solutions for your extensive revision for exams.


Square and Square Roots

In the earlier classes, you have learnt that the area of a square = side x side. That means the area of a square = (side)2 i.e square of side.

If the side of a square is given a units, then

Area of square = (a x a)sq.units = a2 sq.units

Therefore, the square of a number is the product of the number by itself.

In this chapter, you will study about square numbers or perfect squares, their properties and patterns, Pythagorean triplets, various techniques to determine square roots of natural numbers, square roots of decimals and fractions.


Methods of Finding Square Root

  1. By Prime Factorization Method: We can use the prime factorization method to calculate the square root. In this method, first, we need to take out the number and then calculate the prime factors of the number by successive division. Then we need to pair the formulated prime factors in such a way that both the factors are equal in the pair. Determine the product of one factor taken from each pair to obtain the required square root.

  2. By Repeated Subtraction: When the natural numbers are small then we use this method. We keep subtracting 1, 3, 5, 7, 9, …… till we arrive at 0 for calculating the number of odd numbers whose sum is the given natural number. Then we compute the number of times the subtraction is executed to reach at 0. The number of computations becomes the square root of the number.

  3. Long Division Method: We use the long division method when the square numbers are big because the prime factorization method becomes difficult. We draw the bar on each pair of the given number starting from its unit’s place in this method and then divide the digit of the given number at the extreme left by a number whose square root is less than or equal to the leftmost number of the given number.


Write the quotient and the remainder after doing the division. Then we bring forward the next paired digits next to the remainder and continue the division method till you arrive at 0.


The quotient derived in the method becomes the square root of the given number.

Applications of a Square Root

  • If we need to determine the length of the diagonal of a square or a triangle then we use techniques of square roots.

  • Square roots are used when we need to determine the third side of a triangle where the measurement of the other two sides of the triangle is known.

  • Standard deviation can be calculated by using the methods of Square roots.

  • In order to solve a quadratic equation, square roots are used.


Important Facts

Following are the highlights of the chapter that will give you a quick glance of the entire chapter.

  • When the exponent of a natural number is 2, the number derived is called a square number or a perfect square. Ex: 22, 32, 82, etc. are the square numbers or perfect square. 

  • When a natural number is raised to the power 2 then it is called the square of the number.

  • The product of a natural number is called a perfect square when the natural number is multiplied by itself.

  • It is not necessary that all natural numbers are a perfect square.

  • A square number can be defined as the product of equal prime factors.

Ex: 64 = 82 = (2 x 4)2 = 22 x 42.

  • A square number must have 0, 1, 4, 5 or 9 at the unit's place. Opposite of the number does not exist.

  • A natural number can never have 2, 3, 7 or 8 at its unit place then it cannot be a square number.

  • An even number will always have a square number even while an odd number will always have a square number odd.

  • If two consecutive triangle numbers are added, we always obtain a square number.

Ex: 1, 3, 6, 10, …… is the set of triangular numbers and 1 + 3 = 4 = 22.

  • If we take squares of two consecutive natural numbers n & n + 1, we will find 2n non-square natural numbers between them.

  • The total result of the first natural odd natural numbers is n2.

Ex: 1 + 3 + 5 = (sum of first 3 odd natural numbers = 9 = 32)

  • A number can have a perfect square only when the number is not the sum of consecutive odd natural numbers, starting from 1.

  • The sum of two consecutive natural numbers is equal to the square of an odd natural number greater than 1.

i.e. 52 = 25 = 12 + 13, 92 = 81 = 40 + 41, etc.

  • The square of any number having 5 in the unit’s place can be formed by using the formula (a5)2 = a (a + 1) x 100 + 25.

  • A set of three natural numbers a, b, c are said to form a Pythagorean triplet (a, b, c) if  a2 + b2 = c2

  • If n be any natural number m > 1, then (2n, n2 - 1, n2 + 1) will form a Pythagorean triplet.

  • The square root of any natural number, when multiplied by itself, gives the original number n.

  • The square root is the inverse function of finding square. \[\sqrt{\frac{p}{q}}=\frac{\sqrt{p}}{\sqrt{q}}\].

  • There are two important square roots of a perfect square number n + viz. \[\sqrt{n}\] & \[-\sqrt{n}\].

Ex: 62 = 36 = \[\sqrt{36}\] = ±6 i.e. +6 & -6

  • If we know the square of a natural number then we can form the square of the next natural number from the sum of the square, the number & the next natural number.

Ex: 312 = 302 + 30 + 31 = 961.


Benefits of Using Vedantu’s Important Questions

Countless students till today have benefitted from the Important Questions developed by Vedantu. The solutions are designed by very experienced teachers and subject matter experts after extensive research. They have designed the questions and answers in a very simple way by expressing them in a self-explanatory way. Vedantu has provided answers to every question of the chapter and you can revise the answers for your exams. This will uplift your confidence and will also help you with managing time during the exams which are very important. If you have any queries relating to the topic then you can reach out to our teachers on board any time by registering. You can get access to important questions for other chapters and also other subjects. Vedantu is the right platform for you to revise your subjects and improve your marks in exams. You can download the free PDF versions on any device and practice at your own pace. You can also download them from anywhere.


Chapterwise Important Questions for CBSE Class 8 Maths


Also Access, 

 

Conclusion

Now you will be able to master the chapter after going through the above notes and solve the Important Questions for Chapter 6 Maths for Class 8 developed by Vedantu. This is a very important chapter in maths and is the foundation for higher-level maths but there is no need to panic. Vedantu has a panel of very experienced teachers on board. They are from reputed institutions across the country and they can help you overcome your fears for the subject and master the topic.

FAQs on Important Questions for CBSE Class 8 Maths Chapter 6 - Squares and Square Roots

1. What is a square?

The Square of a number is the result obtained when a number is multiplied by itself. For example, the square of 4 is 4 x 4 = 16. If the number is p, then its square is written as p2.

2. Is it possible to obtain the square of a number as a prime number?

No, the square of a number is always a composite number. Even the square of prime numbers is composite numbers. For example: 72 = 49, 112 = 121.

3. Write the square of the first five odd natural numbers.

The first five odd natural numbers are 1, 3, 5, 7 and 9. Their squares are 1, 9, 25, 49 and 81, respectively.

4. Why should we refer to the important questions PDF for Class 8 Maths Chapter 6?

Understanding important concepts, memorising the vital formulae and practising textbook sums form just the tip of the iceberg called exam preparation. Last-minute revision and being prepared with important questions that are likely to feature on the exam paper are the best way to ensure that you are prepared to score high marks. Vedantu’s Important Questions on Class 8 Maths Chapter 6 will help you test your knowledge and gauge your level of preparedness. You will discover your strengths and weaknesses when you work out these questions.