NCERT Solutions & Concepts for Class 8 Maths Chapter 6 Exercise 6.4

Welcome, Class 8 scholars! In this guide, we dive deep into the core of square square roots ex 6 4 class 8 ncert. This exercise is one of the most vital sections of Chapter 6, as it introduces the powerful Long Division Method. While prime factorization works beautifully for smaller numbers, it quickly becomes slow and tedious for very large numbers or numbers with decimals. By mastering the systematic steps of long division, you will easily find the square roots of massive numbers and decimal expressions. We will also explore practical real-world problems, such as finding the least number to add or subtract from a given number to make it a perfect square. Grab your sketchpad, and let us learn these steps together!

Why Use the Long Division Method?

When dealing with large numbers like 2304 or 12544, prime factorization requires finding a long chain of prime factors. Similarly, prime factorization cannot directly solve square roots of decimal numbers like 17.64. The long division method resolves this by grouping digits into pairs (called periods) using bars. We start placing bars from the rightmost digit (units place) for integers. For decimal parts, we group from the decimal point moving left for the integer part, and right for the decimal part. This systematic process helps us find square roots quickly and with absolute certainty.

Step-by-Step Long Division Method

  1. Group the Digits (Placing Bars) — Place a bar over every pair of digits starting from the unit's place. If the total number of digits is odd, the leftmost single digit will also have a bar. Each grouped block under a bar is called a period.
  2. Find the Largest Square — Look at the leftmost period. Find the largest number whose square is less than or equal to this period. Write this number as the divisor and also as the quotient.
  3. Subtract and Bring Down — Subtract the product from the leftmost period. Bring down the next period (pair of digits) to the right of the remainder. This forms your new dividend.
  4. Double the Divisor and Find the Next Digit — Double the value of the quotient and write it with a blank space on its right. Find the largest digit to fill this blank, such that when the new divisor is multiplied by this digit, the product is less than or equal to the current dividend.
  5. Repeat Until Remainder is Zero — Repeat the subtraction and division cycle. If the final remainder is zero, the quotient obtained is the exact square root of the given perfect square.

Worked Examples: Decimals and Perfect Squares

  • Example 1: Finding Square Root of a Decimal (7.29) Step 1: Place bars. The integer part is '7' and decimal part is '29'. Thus, bars are placed as 7.29. Step 2: The largest square less than 7 is 4 (2^2). Write 2 as divisor and quotient. Remainder is 3. Step 3: Put a decimal point in the quotient and bring down '29' to get 329. New divisor starting term is double of 2, which is 4. Step 4: Find a digit 'x' such that 4x x is close to 329. Let us try 7: 47 7 = 329. Perfect! Step 5: So, the square root of 7.29 is 2.7.
  • Example 2: Least Number to Subtract from 400 to make it a perfect square (using a close value like 402) Step 1: Group 402 as 4 02. The largest square for 4 is 2 (22=4). Step 2: Remainder is 0. Bring down 02. Our doubled quotient is 4. The divisor becomes 40 (since 411 = 41 which is > 2). Step 3: 40 * 0 = 0. Subtracting 0 from 2 leaves a remainder of 2. Step 4: Therefore, 2 must be subtracted from 402 to make it a perfect square (402 - 2 = 400, and root of 400 is 20).

Important Exam Tips & Common Mistakes

  1. Decimal Grouping Direction: Never group decimals from right to left! For 15.625, group the integer part as '15' (right to left) and the decimal part as '62' and '50' (left to right). Pad with zeros if necessary.
  2. Addition vs Subtraction Problems: For subtraction word problems, the number to subtract is simply the final remainder. For addition problems, find the square of the next natural number (quotient + 1) and subtract your given number from it. Don't mix up these two methods!
  3. Double the Quotient: Remember to double the entire quotient obtained so far when writing the new divisor, not just the last digit.

Practice Questions with Solutions

  • Q: Find the square root of 529 using the long division method. A: Step 1: Place bars over the pairs. Grouping 529 from right to left gives '5' and '29'. Step 2: The largest number whose square is less than or equal to 5 is 2. Divisor = 2, Quotient = 2. Subtract 4 from 5 to get remainder 1. Step 3: Bring down the next period '29' to make the new dividend 129. Step 4: Double the quotient (2 2 = 4). We need to find a digit to place next to 4. Let's try 3: 43 3 = 129. This matches exactly! Final answer: The square root of 529 is 23.
  • Q: Find the square root of the decimal number 51.84. A: Step 1: Put bars on 51.84. We have two periods: '51' and '84'. Step 2: For 51, the largest square is 49 (7 7). Divisor = 7, Quotient = 7. Remainder = 2. Step 3: Place a decimal in the quotient. Bring down '84' to make the new dividend 284. Step 4: Double the quotient 7 to get 14. Find a digit x such that 14x x is close to 284. Try 2: 142 * 2 = 284. This is an exact match. Final answer: The square root of 51.84 is 7.2.
  • Q: Find the least number that must be subtracted from 3250 so as to get a perfect square. Also find this square root. A: Step 1: Group 3250 as '32' and '50'. Step 2: For 32, the closest square is 25 (5 5). Remainder is 7. Step 3: Bring down '50' to get 750. Double the quotient 5 to get 10. Step 4: Find a digit x such that 10x x <= 750. Try 7: 107 * 7 = 749. Subtract 749 from 750 to get a remainder of 1. Step 5: The remainder is 1. Thus, we must subtract 1 from 3250. Final answer: The least number to subtract is 1. The perfect square is 3249, and its square root is 57.
  • Q: Find the least number that must be added to 525 so as to get a perfect square. Also find this square root. A: Step 1: Group 525 as '5' and '25'. Step 2: For 5, closest square is 4 (2 2). Remainder is 1. Step 3: Bring down '25' to make it 125. Double quotient 2 to get 4. Step 4: Try 42 2 = 84 (remainder 41). This shows 22^2 < 525. Step 5: The next perfect square is 23^2 = 529. The number to be added is 529 - 525 = 4. Final answer: The least number to add is 4. The perfect square is 529, and its square root is 23.

Frequently Asked Questions

What is the difference between subtraction and addition problems in Exercise 6.4?

For subtraction, you find the square root using division and subtract the remainder. For addition, you find the quotient, take the next higher integer, square it, and subtract the original number from this new perfect square.

How do you place bars over decimal numbers?

For decimals, group the whole number part from right to left starting from the decimal point. For the fractional part, group from left to right starting from the decimal point, adding a trailing zero if there is an odd number of digits.

Why is the long division method preferred over prime factorization?

Long division is much faster and simpler for large numbers, non-perfect squares, and decimals where listing prime factors is extremely difficult or mathematically impossible.