Square and Square Roots: A Complete Guide for Class 8 Maths

Understanding Square Numbers and Perfect Squares

A square number (or a perfect square) is the product obtained when an integer is multiplied by itself. For example, when you multiply 3 by 3, you get 9. So, 9 is a square number. We write this as $3 \times 3 = 3^2 = 9$. The small '2' indicates that the number is multiplied by itself.

Think of it visually: if you have a square shape, and its side length is, say, 5 units, then its area is $5 \times 5 = 25$ square units. Here, 25 is a square number. Some common square numbers are:

• $1 \times 1 = 1$ (1 is the square of 1)
• $2 \times 2 = 4$ (4 is the square of 2)
• $4 \times 4 = 16$ (16 is the square of 4)
• $10 \times 10 = 100$ (100 is the square of 10)

An important point to remember is that a natural number is called a perfect square if it is the square of some natural number. For instance, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, etc., are all perfect squares. Numbers like 2, 3, 5, 6, 7, 8, 10, 11, etc., are not perfect squares because they cannot be expressed as the product of an integer with itself. Recognizing perfect squares quickly will be very helpful in your calculations.

Methods to Find Square Roots

1. Method 1: Prime Factorization — This method is useful for finding the square root of perfect squares. It involves breaking down the number into its prime factors. Step 1: Find the prime factorization of the given number. Write the number as a product of its prime factors. Step 2: Group the prime factors in pairs. Since we are looking for a square root, we need factors that appear twice. Step 3: For each pair of identical prime factors, take one factor out. Step 4: Multiply these single factors together. The product is the square root of the original number. Example: Find the square root of 324. Step 1: Prime factorization of 324: $324 = 2 \times 162 = 2 \times 2 \times 81 = 2 \times 2 \times 3 \times 27 = 2 \times 2 \times 3 \times 3 \times 9 = 2 \times 2 \times 3 \times 3 \times 3 \times 3$. Step 2: Group factors in pairs: $(2 \times 2) \times (3 \times 3) \times (3 \times 3)$. Step 3: Take one factor from each pair: $2 \times 3 \times 3$. Step 4: Multiply them: $2 \times 3 \times 3 = 18$. So, $\sqrt{324} = 18$.
2. Method 2: Long Division Method — The long division method is a general way to find the square root of any number, even those that are not perfect squares, to a desired number of decimal places. Step 1: Pair the digits from the right. If it's a whole number, start pairing from the unit's place towards the left. For decimals, pair the digits from the decimal point to the right. If the first group (leftmost) has only one digit, treat it as a pair. Step 2: Find the largest number whose square is less than or equal to the first (leftmost) pair. Write this number as the divisor and the quotient. Subtract its square from the first pair. Step 3: Bring down the next pair of digits to the remainder to form the new dividend. Step 4: Double the current quotient and write it with a blank space next to it. This forms the new partial divisor. Step 5: Find the largest digit to fill the blank space such that the new divisor multiplied by this digit is less than or equal to the new dividend. Write this digit next to the quotient and also in the blank space of the divisor. Step 6: Subtract the product and repeat steps 3-5 until all pairs are brought down or you've reached the desired precision. Example: Find the square root of 529. Step 1: Pair digits: $\overline{5} \ \overline{29}$. (The first pair is 5). Step 2: Largest square $\le 5$ is $2^2=4$. Write 2 as divisor and quotient. $5-4=1$. Step 3: Bring down 29. New dividend is 129. Step 4: Double quotient (2) becomes 4. Place a blank: $4\_$. Step 5: We need $4\_ \times \_ \le 129$. Try 3: $43 \times 3 = 129$. Write 3 in blank and next to quotient. Step 6: Subtract $129-129=0$. All pairs done. $\sqrt{529} = 23$.

Interesting Properties of Square Numbers

• Numbers ending with 2, 3, 7 or 8 are never perfect squares. For example, 12, 23, 37, 48 are not perfect squares. However, numbers ending with 0, 1, 4, 5, 6, 9 can be perfect squares.
• If a number has 0 at its end, its square will have an even number of zeros at the end. For example, $10^2 = 100$ (two zeros), $20^2 = 400$ (two zeros), $100^2 = 10000$ (four zeros).
• The square of an even number is always an even number. Example: $2^2=4$, $4^2=16$, $6^2=36$.
• The square of an odd number is always an odd number. Example: $1^2=1$, $3^2=9$, $5^2=25$.
• For any natural number $n$, the sum of the first $n$ odd natural numbers is $n^2$. Example: $1+3=4=2^2$, $1+3+5=9=3^2$.

Practice Questions with Solutions

• Q: Is 256 a perfect square? If yes, find its square root using prime factorization. A: Step 1: Find the prime factorization of 256. $256 = 2 \times 128 = 2 \times 2 \times 64 = 2 \times 2 \times 2 \times 32 = 2 \times 2 \times 2 \times 2 \times 16 = 2 \times 2 \times 2 \times 2 \times 2 \times 8 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 4 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$ So, $256 = 2^8$. Step 2: Group the prime factors in pairs. $256 = (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2)$ Step 3: Take one factor from each pair. $2 \times 2 \times 2 \times 2$ Step 4: Multiply these factors. $2 \times 2 \times 2 \times 2 = 16$ Final answer: Yes, 256 is a perfect square, and its square root is 16.
• Q: Find the square root of 1024 using the long division method. A: Step 1: Pair the digits from the right: $\overline{10} \ \overline{24}$. Step 2: For the first pair (10), the largest square less than or equal to 10 is $3^2=9$. Write 3 as divisor and quotient. Subtract $10-9=1$. Step 3: Bring down the next pair (24). New dividend is 124. Step 4: Double the quotient (3) to get 6. Add a blank space: $6\_$. Step 5: We need $6\_ \times \_ \le 124$. Try 2: $62 \times 2 = 124$. Write 2 in the blank and next to the quotient. Step 6: Subtract $124-124=0$. All pairs are done. Final answer: The square root of 1024 is 32.
• Q: What is the smallest number by which 72 must be multiplied to make it a perfect square? A: Step 1: Find the prime factorization of 72. $72 = 2 \times 36 = 2 \times 2 \times 18 = 2 \times 2 \times 2 \times 9 = 2 \times 2 \times 2 \times 3 \times 3$ Step 2: Group the prime factors in pairs: $(2 \times 2) \times (3 \times 3) \times 2$. Step 3: Identify the factor that is not in a pair. Here, '2' is not in a pair. Step 4: To make 72 a perfect square, we need to multiply it by the missing factor to complete its pair. We need one more '2'. So, multiply by 2. $72 \times 2 = 144$. And $\sqrt{144} = 12$. Final answer: The smallest number is 2.
• Q: Estimate the square root of 600 to the nearest whole number. A: Step 1: Find perfect squares just below and just above 600. We know $20^2 = 400$ and $30^2 = 900$. Let's try values between 20 and 30. $24^2 = 576$ $25^2 = 625$ Step 2: Compare 600 with the perfect squares found. 600 is between 576 and 625. Step 3: Determine which perfect square 600 is closer to. Difference from 576: $600 - 576 = 24$ Difference from 625: $625 - 600 = 25$ 600 is closer to 576. Final answer: The estimated square root of 600 to the nearest whole number is 24.

Common Mistakes and How to Avoid Them

Many students make small errors that can cost marks. Here are a few common mistakes and tips to avoid them:

1. Confusing Square with Twice: Remember, $5^2$ means $5 \times 5 = 25$, NOT $5 \times 2 = 10$. Always multiply the number by itself.
2. Incorrect Pairing in Prime Factorization: When finding the square root, make sure you group identical prime factors in pairs. If a factor is left unpaired, the number is not a perfect square, or you need to adjust it (e.g., multiply/divide).
3. Errors in Long Division Steps: The long division method requires careful calculation. A common mistake is not doubling the quotient correctly in each step or choosing a wrong digit for the divisor. Double-check your multiplication and subtraction in each step.
4. Forgetting Decimal Points: When finding the square root of a decimal number using long division, remember to place the decimal point in the quotient exactly above the decimal point in the number you are dividing. Also, pairs of digits are formed from the decimal point outwards.

Frequently Asked Questions