Square and Square Roots: Exercise 6.3 Guide

Welcome, brilliant minds! In this chapter on Squares and Square Roots, we're diving deep into NCERT Exercise 6.3. Think of a square root as a number's 'root' or origin. If 4 × 4 = 16, then 4 is the square root of 16. This exercise is all about becoming a detective and finding these roots! We will master two powerful methods: finding square roots by repeated subtraction and the super useful prime factorization method. You'll also learn cool tricks, like figuring out the possible last digit of a square root just by looking at the last digit of the number itself. By the end of this session, you'll be able to find the square root of any perfect square and even figure out what's missing from a number to make it a perfect square. Let's start investigating!

The Unit Digit Clue for Square Roots

Did you know that the last digit of a number can give you a big clue about its square root? This is a fantastic shortcut for many problems in Exercise 6.3. When you square a number, the unit digit of the result depends only on the unit digit of the original number. Let's see the pattern:

If a number ends in 1 (like 81), its square root must end in 1 (since 1²=1) or 9 (since 9²=81).
If a number ends in 4 (like 64), its square root must end in 2 (since 2²=4) or 8 (since 8²=64).
If a number ends in 9 (like 49), its square root must end in 3 (since 3²=9) or 7 (since 7²=49).
If a number ends in 6 (like 36), its square root must end in 4 (since 4²=16) or 6 (since 6²=36).
If a number ends in 5 (like 25), its square root must end in 5.
If a number ends in 0 (like 100), its square root must end in 0.

Notice something important? No perfect square ever ends in 2, 3, 7, or 8. This is a quick way to identify numbers that are not perfect squares.

How to Find Square Roots Using Prime Factorization

Step 1: Find Prime Factors — Break down the given number into its smallest prime factors. You can do this using the division method. For example, to find the prime factors of 324, you would do: 324 ÷ 2 = 162 162 ÷ 2 = 81 81 ÷ 3 = 27 27 ÷ 3 = 9 9 ÷ 3 = 3 3 ÷ 3 = 1 So, 324 = 2 × 2 × 3 × 3 × 3 × 3.
Step 2: Make Pairs of Identical Factors — Group the prime factors into pairs of two. For 324, the pairs are (2 × 2) and (3 × 3) and (3 × 3). If you can't make complete pairs (i.e., a factor is left over), the number is not a perfect square.
Step 3: Pick One Factor from Each Pair — For every pair you've made, take only one of the numbers. From our example, we take one '2' from the first pair, one '3' from the second pair, and another '3' from the third pair.
Step 4: Multiply to Get the Square Root — Multiply all the factors you picked in the previous step. For our example, this would be 2 × 3 × 3 = 18. This final product is the square root of the original number. So, the square root of 324 is 18.

Worked Examples: Applying the Methods

Example 1: Find the square root of 1764 by prime factorization. Step 1: Find the prime factors of 1764. 1764 = 2 × 882 = 2 × 2 × 441 = 2 × 2 × 3 × 147 = 2 × 2 × 3 × 3 × 49 = 2 × 2 × 3 × 3 × 7 × 7 Step 2: Pair the prime factors: (2 × 2) × (3 × 3) × (7 × 7). Step 3: Take one factor from each pair: 2, 3, and 7. Step 4: Multiply these factors: 2 × 3 × 7 = 42. Answer: The square root of 1764 is 42.
Example 2: Find the smallest whole number to multiply with 180 to make it a perfect square. Step 1: Prime factorize 180. We get 180 = 2 × 90 = 2 × 2 × 45 = 2 × 2 × 3 × 15 = 2 × 2 × 3 × 3 × 5. Step 2: Identify unpaired factors by making pairs: (2 × 2) × (3 × 3) × 5. Step 3: The factor '5' is left without a pair. To make 180 a perfect square, we need to complete the pair for 5. This means we must multiply the original number by another 5. Answer: The smallest whole number to multiply by is 5. The new number will be 180 × 5 = 900, and its square root will be 2 × 3 × 5 = 30.

Important Points to Remember

Not a Perfect Square: If a prime factor is left over after pairing, the number is not a perfect square.
Quick Elimination: Numbers ending in 2, 3, 7, or 8 are never perfect squares. This is a very useful check!
Watch the Zeros: A number ending in an odd number of zeros (like 10, 1000, 500) is never a perfect square. Perfect squares must have an even number of zeros at the end (like 100, 90000).
Multiply vs. Divide: When asked to find the smallest number to multiply to make a perfect square, you supply the missing factor for the pair. When asked to divide, you remove the unpaired factor.

Practice Questions with Solutions

Q: What could be the possible ‘ones’ digit of the square root of 65766? A: Step 1: Identify the unit digit of the given number. The number is 65766, and its unit digit is 6. Step 2: Recall which squares result in a unit digit of 6. We know that 4² = 16 and 6² = 36. Step 3: Therefore, the unit digit of the square root of a number ending in 6 must be either 4 or 6. Final answer: The possible ones digits are 4 or 6.
Q: Find the square root of 4096 by the prime factorization method. A: Step 1: Find the prime factors of 4096. 4096 = 2 × 2048 = 2 × 2 × 1024 = 2 × 2 × 2 × 512 = 2 × 2 × 2 × 2 × 256 = ... continuing this process, we find that 4096 = 2¹². So, 4096 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2. Step 2: Group the factors into pairs. We get six pairs of 2s: (2×2) × (2×2) × (2×2) × (2×2) × (2×2) × (2×2). Step 3: Take one factor from each pair. We get six 2s: 2, 2, 2, 2, 2, 2. Step 4: Multiply these factors: 2 × 2 × 2 × 2 × 2 × 2 = 64. Final answer: The square root of 4096 is 64.
Q: Find the smallest whole number by which 1008 should be divided to get a perfect square number. Also, find the square root of the number so obtained. A: Step 1: Find the prime factors of 1008. 1008 = 2 × 504 = 2 × 2 × 252 = 2 × 2 × 2 × 126 = 2 × 2 × 2 × 2 × 63 = 2 × 2 × 2 × 2 × 3 × 21 = 2 × 2 × 2 × 2 × 3 × 3 × 7. Step 2: Make pairs: (2×2) × (2×2) × (3×3) × 7. The factor 7 is left unpaired. Step 3: To make the number a perfect square, we must divide by the unpaired factor to remove it. So, we must divide 1008 by 7. Step 4: The new number is 1008 ÷ 7 = 144. The prime factors of 144 are (2×2) × (2×2) × (3×3). The square root is 2 × 2 × 3 = 12. Final answer: The smallest whole number to divide by is 7. The resulting perfect square is 144, and its square root is 12.
Q: 2025 plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows. Find the number of rows and the number of plants in each row. A: Step 1: Understand the problem. Let the number of rows be 'x'. The problem states that the number of plants in each row is also 'x'. The total number of plants is the number of rows multiplied by the plants per row, which is x × x = x². Step 2: We are given that the total number of plants is 2025. So, x² = 2025. We need to find x, which is the square root of 2025. Step 3: Find the square root of 2025 using prime factorization. 2025 = 3 × 675 = 3 × 3 × 225 = 3 × 3 × 3 × 75 = 3 × 3 × 3 × 3 × 25 = 3 × 3 × 3 × 3 × 5 × 5. Step 4: Make pairs: (3×3) × (3×3) × (5×5). Take one from each pair and multiply: 3 × 3 × 5 = 45. Final answer: There are 45 rows, and each row has 45 plants.

Frequently Asked Questions

How can I quickly tell if a number is NOT a perfect square?

Look at the digit in the units place. If the number ends in 2, 3, 7, or 8, it can never be a perfect square. Also, if it ends in an odd number of zeros (like 500 or 7000), it's not a perfect square.

What is the difference between finding a square and a square root?

They are opposite operations. Finding the square of a number means multiplying it by itself (e.g., the square of 5 is 5×5=25). Finding the square root means discovering which number, when multiplied by itself, gives the original number (e.g., the square root of 25 is 5).

Why does the prime factorization method work for finding square roots?

When you square a number (like 12), you are squaring its prime factors (2×2×3)² = (2×2×3)×(2×2×3). Notice every factor now has a pair. Finding the square root by prime factorization is simply this process in reverse: you group the factors into pairs and take one from each pair to find the original 'root' number.