Squares and Square Roots: CBSE Class 8 Maths

What is a Square Number?

A square number, also known as a perfect square, is what you get when you multiply an integer (a whole number) by itself. For example, when we multiply 5 by itself (5 × 5), we get 25. So, 25 is a square number. We write this as 5² = 25, and we read it as '5 squared equals 25'.

Think of it visually. If you have 9 marbles, can you arrange them into a solid square shape? Yes! You can make a 3×3 square. So, 9 is a perfect square (3²). But what if you have 12 marbles? You can't arrange them into a solid square. You could make a 3×4 rectangle, but not a square. This means 12 is not a perfect square.

The first few square numbers are:
1 × 1 = 1 (1²)
2 × 2 = 4 (2²)
3 × 3 = 9 (3²)
4 × 4 = 16 (4²)
5 × 5 = 25 (5²)
...and so on. Understanding this simple idea is the first step to mastering the entire chapter.

Important Properties of Square Numbers

• A number ending in 2, 3, 7, or 8 can never be a perfect square. For example, 32, 53, and 108 are not perfect squares.
• A perfect square can only end with the digits 0, 1, 4, 5, 6, or 9.
• The number of zeros at the end of a perfect square is always even. For instance, 100 (2 zeros) is a perfect square (10²), but 1000 (3 zeros) is not.
• The square of an even number is always an even number. (e.g., 6² = 36)
• The square of an odd number is always an odd number. (e.g., 7² = 49)

How to Find Square Roots: Prime Factorisation Method

1. Step 1: Find the Prime Factors — Start by breaking down the number into its prime factors. Let's find the square root of 784. We do this by repeated division: 784 ÷ 2 = 392 392 ÷ 2 = 196 196 ÷ 2 = 98 98 ÷ 2 = 49 49 ÷ 7 = 7 7 ÷ 7 = 1 So, the prime factors of 784 are 2 × 2 × 2 × 2 × 7 × 7.
2. Step 2: Group the Factors into Pairs — Now, group the identical prime factors into pairs. For 784, the factors are (2 × 2) × (2 × 2) × (7 × 7).
3. Step 3: Take One Factor from Each Pair — From each pair of identical factors, pick just one number. From (2 × 2), we take one 2. From the next (2 × 2), we take another 2. From (7 × 7), we take one 7.
4. Step 4: Multiply the Chosen Factors — Finally, multiply the numbers you picked in the previous step. In our case, this is 2 × 2 × 7 = 28. Therefore, the square root of 784 is 28. We write this as √784 = 28. You can check this by calculating 28 × 28, which equals 784.

Worked Example: A Real-World Problem

• Problem: A school principal wants to arrange 625 students in the assembly ground in such a way that the number of rows is equal to the number of columns. How many students would be in each row? Step 1: Understand the problem. The arrangement of students in equal rows and columns forms a square. The total number of students (625) is the area of this square. We need to find the number of students in one row, which is the side length of the square, or the square root of 625. Step 2: Find the square root of 625 using prime factorisation. 625 ÷ 5 = 125 125 ÷ 5 = 25 25 ÷ 5 = 5 5 ÷ 5 = 1 So, 625 = 5 × 5 × 5 × 5. Step 3: Pair the factors. 625 = (5 × 5) × (5 × 5). Step 4: Take one factor from each pair and multiply. From the first pair, we take a 5. From the second pair, we take another 5. The square root is 5 × 5 = 25. Final Answer: There would be 25 students in each row.

Practice Questions with Solutions

• Q: Find the square of 35. A: Step 1: The square of a number is the number multiplied by itself. Step 2: Calculate 35 × 35. 35 × 35 = 1225. Final answer: The square of 35 is 1225.
• Q: Is 240 a perfect square? Why or why not? A: Step 1: Recall the properties of perfect squares. A number ending in an odd number of zeros is not a perfect square. Step 2: Observe the number 240. It ends in a single zero (which is an odd number of zeros). Step 3: Alternatively, check the prime factors of 240: 240 = 2 × 2 × 2 × 2 × 3 × 5. The factors 3 and 5 are not in pairs. Final answer: No, 240 is not a perfect square because its prime factors cannot all be grouped into identical pairs.
• Q: Find the smallest whole number by which 180 should be multiplied to get a perfect square number. A: Step 1: Find the prime factors of 180. 180 = 2 × 90 = 2 × 2 × 45 = 2 × 2 × 3 × 15 = 2 × 2 × 3 × 3 × 5. Step 2: Group the factors into pairs: (2 × 2) × (3 × 3) × 5. Step 3: Identify the unpaired factor. The factor 5 is left without a pair. Step 4: To make it a perfect square, we need to multiply by another 5 to create a pair. Final answer: The smallest whole number to multiply by is 5.
• Q: Find the square root of 441 using the prime factorisation method. A: Step 1: Find the prime factors of 441. Since it's an odd number, we start with 3. Sum of digits 4+4+1=9, which is divisible by 3. 441 ÷ 3 = 147 147 ÷ 3 = 49 49 ÷ 7 = 7 7 ÷ 7 = 1 So, 441 = 3 × 3 × 7 × 7. Step 2: Group the prime factors into pairs: (3 × 3) × (7 × 7). Step 3: Take one factor from each pair and multiply them. We take one 3 from the first pair and one 7 from the second pair. √441 = 3 × 7 = 21. Final answer: The square root of 441 is 21.

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