CBSE Class 8 Maths: Squares and Square Roots Ex 6.2

Welcome, Class 8 students, to an exciting journey into the world of "Squares and Square Roots"! In this chapter, you'll discover a special type of number called a square number and learn how to find its square root. Specifically, Exercise 6.2 focuses on finding the squares of numbers efficiently, especially by using algebraic identities like $(a+b)^2$ and $(a-b)^2$. Understanding these methods will not only make calculations faster but also build a strong foundation for more advanced algebra. By the end of this page, you'll be able to confidently calculate squares of numbers, understand the patterns involved, and apply useful mathematical techniques. Let's unlock the secrets of squares together and ace your CBSE Class 8 Maths exams!

Understanding Square Numbers and Their Properties

A square number, also known as a perfect square, is the result of multiplying an integer by itself. For example, when you multiply 3 by 3, you get 9. So, 9 is a square number. We write this as $3 \times 3 = 3^2 = 9$. Similarly, $5 \times 5 = 5^2 = 25$, making 25 a square number. Think of it like finding the area of a square! If a square has a side length of 's' units, its area is $s \times s = s^2$ square units.

The inverse operation of squaring a number is finding its square root. The square root of 9 is 3, because $3^2 = 9$. We use the symbol $\sqrt{}$ for square root. So, $\sqrt{9} = 3$. In Exercise 6.2, we'll focus on how to efficiently calculate the square of a number, especially larger ones, without just multiplying it out directly. This involves using some clever algebraic identities that simplify the process.

Finding Squares Using Algebraic Identities

  1. Step 1: Break Down the Number — To use identities, we need to express the given number as a sum or difference of two convenient numbers. Usually, one of these numbers is a multiple of 10. For example, to find the square of 26, we can write it as $(20 + 6)$. To find the square of 48, we can write it as $(50 - 2)$. Choosing numbers that are easy to square (like 20, 50) makes the calculation simpler.
  2. Step 2: Choose the Correct Identity — Once you've broken down the number, decide whether it fits the form $(a+b)$ or $(a-b)$. If the number is expressed as a sum, like $(20+6)$, use the identity: $(a+b)^2 = a^2 + 2ab + b^2$. If the number is expressed as a difference, like $(50-2)$, use the identity: $(a-b)^2 = a^2 - 2ab + b^2$.
  3. Step 3: Substitute and Calculate — Substitute the values of 'a' and 'b' into the chosen identity. Then, perform the multiplications and additions/subtractions carefully. Remember to calculate $a^2$, $2ab$, and $b^2$ separately before combining them. For instance, if you have $(20+6)^2$, then $a=20$ and $b=6$. You would calculate $20^2$, $2 \times 20 \times 6$, and $6^2$, and then add them all together.
  4. Step 4: Final Summation — Add or subtract the resulting terms to get the final square of the number. Ensure your calculations are precise to avoid errors. This method is much faster than direct multiplication for larger numbers and helps in understanding algebraic principles.

Worked Examples: Calculating Squares Using Identities

  • Example 1: Find the square of 26. Step 1: Express 26 as a sum. $26 = 20 + 6$. Step 2: Apply the identity $(a+b)^2 = a^2 + 2ab + b^2$ with $a=20$ and $b=6$. Step 3: Substitute the values: $(20+6)^2 = (20)^2 + (2 \times 20 \times 6) + (6)^2$ $= 400 + 240 + 36$ Step 4: Add the terms: $400 + 240 + 36 = 676$. Final Answer: $26^2 = 676$.
  • Example 2: Calculate $(32)^2$. Step 1: Express 32 as a sum. $32 = 30 + 2$. Step 2: Apply the identity $(a+b)^2 = a^2 + 2ab + b^2$ with $a=30$ and $b=2$. Step 3: Substitute the values: $(30+2)^2 = (30)^2 + (2 \times 30 \times 2) + (2)^2$ $= 900 + 120 + 4$ Step 4: Add the terms: $900 + 120 + 4 = 1024$. Final Answer: $(32)^2 = 1024$.
  • Example 3: Find the square of 49. Step 1: Express 49 as a difference. $49 = 50 - 1$. Step 2: Apply the identity $(a-b)^2 = a^2 - 2ab + b^2$ with $a=50$ and $b=1$. Step 3: Substitute the values: $(50-1)^2 = (50)^2 - (2 \times 50 \times 1) + (1)^2$ $= 2500 - 100 + 1$ Step 4: Perform operations: $2400 + 1 = 2401$. Final Answer: $49^2 = 2401$.
  • Example 4: Calculate $(88)^2$. Step 1: Express 88 as a difference. $88 = 90 - 2$. Step 2: Apply the identity $(a-b)^2 = a^2 - 2ab + b^2$ with $a=90$ and $b=2$. Step 3: Substitute the values: $(90-2)^2 = (90)^2 - (2 \times 90 \times 2) + (2)^2$ $= 8100 - 360 + 4$ Step 4: Perform operations: $7740 + 4 = 7744$. Final Answer: $(88)^2 = 7744$.

Exam Tip: Mastering Identities for Squares

To excel in questions related to finding squares using identities, remember these key points:

  1. Memorize Identities: Ensure you know $(a+b)^2 = a^2 + 2ab + b^2$ and $(a-b)^2 = a^2 - 2ab + b^2$ by heart. A common mistake is forgetting the '$2ab
    term or using the wrong sign for it in the $(a-b)^2$ identity.
  2. Strategic Breaking Down: Always try to break the number into a sum or difference involving a multiple of 10 (e.g., 20, 30, 50, 100). This simplifies the calculation of $a^2$ and $2ab$. For example, for 39, $(40-1)$ is easier than $(30+9)$.
  3. Careful with Signs: Pay close attention to the signs, especially when using $(a-b)^2$. The middle term '$2ab
    is subtracted.
  4. Quick Check: For a small sanity check, look at the last digit of your answer. The last digit of a square number is determined by the last digit of the original number (e.g., if a number ends in 6, its square must end in $6^2=36$, so 6).

Practice Questions with Solutions

  • Q: Find the square of 39 using an identity. A: Step 1: Express 39 as a difference: $39 = 40 - 1$. Step 2: Apply the identity $(a-b)^2 = a^2 - 2ab + b^2$ with $a=40$ and $b=1$. Step 3: Substitute values: $(40-1)^2 = (40)^2 - (2 \times 40 \times 1) + (1)^2 = 1600 - 80 + 1$. Step 4: Calculate: $1520 + 1 = 1521$. Final answer: $39^2 = 1521$.
  • Q: Calculate $(71)^2$ using a suitable identity. A: Step 1: Express 71 as a sum: $71 = 70 + 1$. Step 2: Apply the identity $(a+b)^2 = a^2 + 2ab + b^2$ with $a=70$ and $b=1$. Step 3: Substitute values: $(70+1)^2 = (70)^2 + (2 \times 70 \times 1) + (1)^2 = 4900 + 140 + 1$. Step 4: Calculate: $5040 + 1 = 5041$. Final answer: $(71)^2 = 5041$.
  • Q: What is the value of $(98)^2$? A: Step 1: Express 98 as a difference: $98 = 100 - 2$. Step 2: Apply the identity $(a-b)^2 = a^2 - 2ab + b^2$ with $a=100$ and $b=2$. Step 3: Substitute values: $(100-2)^2 = (100)^2 - (2 \times 100 \times 2) + (2)^2 = 10000 - 400 + 4$. Step 4: Calculate: $9600 + 4 = 9604$. Final answer: $(98)^2 = 9604$.
  • Q: Find the square of 54. A: Step 1: Express 54 as a sum: $54 = 50 + 4$. Step 2: Apply the identity $(a+b)^2 = a^2 + 2ab + b^2$ with $a=50$ and $b=4$. Step 3: Substitute values: $(50+4)^2 = (50)^2 + (2 \times 50 \times 4) + (4)^2 = 2500 + 400 + 16$. Step 4: Calculate: $2900 + 16 = 2916$. Final answer: $54^2 = 2916$.

Frequently Asked Questions

What is a square number?

A square number is the result of multiplying an integer by itself. For instance, $4 \times 4 = 16$, so 16 is a square number. It's often written with a small '2' above the number, like $4^2$.

Why do we use identities to find squares?

Using algebraic identities like $(a+b)^2$ or $(a-b)^2$ simplifies the process of finding squares for larger numbers. Instead of long multiplication, you break the number into simpler parts, perform easier calculations, and then combine them, which is faster and less prone to errors.

Can I always use either $(a+b)^2$ or $(a-b)^2$?

Yes, you can choose whichever identity makes the calculation easiest. For example, for 49, $(50-1)^2$ is often simpler than $(40+9)^2$. The goal is to pick 'a' and 'b' values that are easy to square and multiply.