CBSE Notes Class 8 Maths Chapter 5 Squares and Square Roots

1.0Introduction to Square Numbers

Squares and Square Roots is an essential part of the Maths curriculum. Chapter 5 notes examine the detailed ideas of squares and their square roots, giving students all the essential knowledge and abilities. Here is the basic understanding of what you will study in this chapter.

• A whole number that can be expressed as the product of an integer and itself is called a square number. Specifically, n × n = n²
• Among the square numbers are 1, 4, 9, 16, and 25.

2.0CBSE Class 8 Maths Notes Chapter 5: Squares and Square Roots - Revision Notes

Properties of Square Numbers

• Perfect squares have their last digits only limited to the values 0, 1, 4, 5, 6, or 9. 
• The last digits of 2, 3, 7, and 8 are not perfect squares.
• An odd number's square is always an odd number, while an even number's square is always an even number.
• The number of zeros at the result of a perfect square is always even, and it is always twice as many as the original number.

Interesting Patterns

• When two consecutive triangular numbers are added together, the result is a square number.
• The squares of two consecutive natural numbers, n and n + 1, with 2n non-perfect square numbers in between, are known as numbers between square numbers.
• Adding Odd Numbers: The sum of successive odd numbers, beginning with 1, can be used to calculate a number's square.
• Product of Consecutive Even or Odd integers: A perfect square is always produced when two consecutive even or odd integers are multiplied.

Finding the Square of a Number

• The formula n² provides the square of a number n.
• The square of an integer may be found using a variety of identities and shortcuts, including (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b².

Square Roots

• A number's square root is defined as the factor of that number, which, when combined with itself, returns the original number.
• The square root of an integer is represented mathematically by the symbol "√." 
• The square root of a perfect square yields an integer; however, the square root of an imperfect square is a non-terminating decimal.

Methods to Find Square Roots

• Repeated Subtraction: Subtract odd numbers repeatedly, starting with one and ending with 0, to determine the square root of a number.
• Prime Factorisation: By factoring a number into its prime components and then calculating the product of the factors that occur in pairs.
• Using the Long Division Approach: Which divides a number into pairs of digits and then finds the greatest digit that, when multiplied by itself, is less than or equal to the current pair of digits.

Estimating Square Roots

• To estimate the square root of non-perfect square values, find the nearest perfect square and then adjust the value based on the difference between the given number and the perfect square. It can be illustrated by the following example.

Estimate the square root of 300.

We know that, 100 < 300 < 400.

Since, and

So, 10 < < 20 

We know that, 17² = 289 and 18² = 324.

Thus, 17 < < 18.

But, 300 is closer to 289 as compared to 324.

Therefore, is approximately equal to 17.

Finding square root through prime factorisation

In order to find the square root of a perfect square by prime factorisation, we follow the following steps.

(i) Obtain the given number.

(ii) Resolve the given number into prime factors by successive division.

(iii) Make pairs of prime factors such that both the factors in each pair are equal.

(iv) Take one factor from each pair and find their product.

(v) The product obtained is the required square root.

Pythagorean Triplets

Consider the following, 3² + 4² = 9 + 16 = 25 = 5²

The collection of numbers 3, 4 and 5 is known as Pythagorean triplet.

6, 8, 10 is also a Pythagorean triplet 

Since, 6² + 8² = 36 + 64 = 100 = 10²

Generalising it, for any natural number m > 1, 

we have, (2m)² + (m² – 1)² = (m² + 1)²

So, 2m, m² – 1 and m² + 1 form a Pythagorean triplet.

3.0Key Features of CBSE Class 8 Maths Notes for Chapter 5: Squares and Square Roots

1. Step-by-Step Explanations: To help students understand the fundamental ideas, the notes include thorough and precise explanations of every subject, along with step-by-step examples.
2. Practical Applications: By highlighting the principles discussed in the chapter's real-world applications, the notes help students appreciate the significance and relevance of the subjects.
3. Revision-Friendly: The notes' clear and organised structure makes them a great tool for speedy review and test preparation.

4.0Solved Questions of CBSE Class 8 Maths Notes for Chapter 5: Squares and Square Roots

Exploring Square Numbers and Their Properties

Problem 1: Unit Digit Detective Let's explore how the unit digit of a number changes when squared!

i. Square of 81 Challenge: What will be the unit digit when 81 is squared? 

Solution:

• Look at the unit's place digit: 1
• A key rule: Numbers ending in 1 always have a square ending in 1
• Verification: 81² = 6,561 ✓ (Ends in 1!)

ii. Square of 272 Challenge: Predict the unit digit of 272 when squared Solution:

• Unit's place digit: 2
• Another fun rule: Numbers ending in 2 always square to a number ending in 4
• Verification: 272² = 73,984 ✓ (Ends in 4!)

Problem 2: Spotting Non-Perfect Squares Can you identify numbers that cannot be perfect squares?

i. Is 1057 a Perfect Square? 

Solution:

• Perfect squares can only end in: 0, 1, 4, 5, 6, or 9
• 1057 ends in 7
• Verdict: NOT a perfect square!

ii. Is 23453 a Perfect Square? Solution:

• Perfect squares can only end in: 0, 1, 4, 5, 6, or 9
• 23453 ends in 3
• Verdict: NOT a perfect square!

Problem 3: We will look at an intriguing pattern in square numbers.

Pattern Challenge:

• 11² = 121
• 101² = 10,201
• 1001² = 1,002,001
• 100,001² = 10,000,200,001

What's the pattern for 10,000,001²?

Solution:

Take note of the pattern: There are the same number of zeroes before and after the middle digit as there were in the original number.

7 digits for 10,000,001

Therefore, there will be three zeroes before and after the middle number in 10,000,001².

The outcome was 100,000,020,000,001

Problem 4: Find a Pythagorean triplet where one of the numbers is 12.

Solution:

If we take m² – 1 = 12

⇒ m² = 12 + 1 = 13

then the value of m will not be an integer.

So, we try to take, m² + 1 = 12

⇒ m² = 12 – 1 = 11

Again, the value of m will not be an integer.

So, let us take 2m = 12

⇒ m = 6

Thus, the other members of Pythagorean triplet are as follows 

m² – 1 = 6² – 1 = 36 – 1 = 35

m² + 1 = 6² + 1 = 36 + 1 = 37

Thus, the required triplet is 12, 35 and 37.

5.0Sample Questions

Q1. What is a common mistake for square roots?

Answer: One common error students make while calculating square roots is to interchange the square root symbol $(\sqrt{})$ with the radical sign $(\sqrt[3]{)})$, which represents the cube root.

Q2. Which formulae and shortcuts are discussed in Chapter 5 of the CBSE Class 8 Maths Notes?

Answer: The notes include several shortcuts and formulas, among which is the explanation of how to calculate the square of a number using (a + b)² = a² + 2ab + b² and (a – b)² = a² – 2ab + b², the addition of triangle numbers, the rationale behind the square of any odd number is the addition of n consecutive odd numbers, as well as the techniques of finding the square root by using repeated subtraction, prime factorisation, and long division methods.