Squares And Square Roots Class 8 Chapter Notes

Key Terms and Definitions

Square Number

The number obtained by multiplying a number by itself. For a number 'x', its square is x² (x * x).

Perfect Square

A natural number that is the square of another natural number. For example, 36 is a perfect square because 36 = 6².

Square Root

A number which, when multiplied by itself, gives the original number. The symbol for square root is √. For example, √36 = 6.

Prime Factorization Method

A method to find the square root of a number by expressing it as a product of its prime factors and then pairing them.

Long Division Method

A step-by-step method to find the square root of large numbers, numbers that are not perfect squares, and decimals.

Pythagorean Triplet

A set of three positive integers (a, b, c) that satisfy the equation a² + b² = c². The most common example is (3, 4, 5).

Properties of Square Numbers

Understanding the properties of square numbers can save a lot of time in calculations and help you quickly identify whether a number can be a perfect square. A key observation is related to the unit digit (the digit in the one's place). A perfect square can only end with the digits 0, 1, 4, 5, 6, or 9. This means any number ending in 2, 3, 7, or 8 is never a perfect square. Another property involves zeros. A number ending in an odd number of zeros is never a perfect square. For example, 100 (two zeros) is a perfect square (10²), but 1000 (three zeros) is not. Furthermore, the square of an even number is always even (e.g., 4² = 16), and the square of an odd number is always odd (e.g., 5² = 25). There are also interesting patterns between consecutive squares. The difference between the squares of two consecutive natural numbers, n and (n+1), is equal to their sum: (n+1)² - n² = (n+1) + n = 2n + 1.

Must Remember Facts

• Numbers ending in 2, 3, 7, or 8 are never perfect squares.
• Perfect squares can only have an even number of zeros at the end.
• The square of an even number is always even, and the square of an odd number is always odd.
• For any natural number m > 1, (2m, m²-1, m²+1) forms a Pythagorean triplet.
• The number of non-perfect square numbers between the squares of n and (n+1) is 2n.
• If a perfect square has 'n' digits, its square root will have n/2 digits (if n is even) or (n+1)/2 digits (if n is odd).
• The sum of the first 'n' odd natural numbers is n². For example, 1 + 3 + 5 = 9 = 3².
• A number is a perfect square if all its prime factors can be grouped into pairs.

How to Find Square Root by Long Division Method

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Worked Examples

• {"title":"Example 1: Find the square root of 784 by prime factorization.","bodyMarkdown":"Solution:\n1. Find the prime factors of 784.\n784 = 2 x 392 = 2 x 2 x 196 = 2 x 2 x 2 x 98 = 2 x 2 x 2 x 2 x 49 = 2 x 2 x 2 x 2 x 7 x 7\n2. Group the factors into pairs: 784 = (2 x 2) x (2 x 2) x (7 x 7)\n3. Take one factor from each pair: √784 = 2 x 2 x 7 = 28"}
• {"title":"Example 2: Find a Pythagorean triplet whose smallest member is 8.","bodyMarkdown":"Solution:\n1. The general form of a Pythagorean triplet is (2m, m²-1, m²+1).\n2. Let 2m = 8. This gives m = 4.\n3. Now find the other two members:\nm²-1 = 4² - 1 = 16 - 1 = 15\nm²+1 = 4² + 1 = 16 + 1 = 17\n4. The Pythagorean triplet is (8, 15, 17).\nCheck: 8² + 15² = 64 + 225 = 289, and 17² = 289."}
• {"title":"Example 3: Find the smallest whole number by which 252 should be multiplied to get a perfect square.","bodyMarkdown":"Solution:\n1. Prime factorize 252: 252 = 2 x 2 x 3 x 3 x 7\n2. Group the factors: 252 = (2 x 2) x (3 x 3) x 7\n3. The prime factor 7 is left unpaired.\n4. To make it a perfect square, we need to multiply by 7 to complete the pair.\n5. The smallest whole number is 7."}

Exam Tips for Squares and Square Roots

In your exam, speed and accuracy are key. Memorize the squares of numbers from 1 to 25. This will help you quickly identify perfect squares and their roots without calculation. When using the long division method for decimals like √12.96, remember to place the decimal point in the quotient as soon as you bring down the first pair of digits after the decimal. A common error is mismanaging pairs. For whole numbers, pair from the right (e.g., 1764 becomes 17 64). For the decimal part, pair from the left (e.g., 0.8281 becomes .82 81).

Practice Questions with Solutions

• What is the unit digit of the square of the number 854? The unit digit is 6, because the unit digit of 854 is 4, and 4² = 16, which has a unit digit of 6.
• How many non-square numbers lie between 12² and 13²? The number of non-square numbers between n² and (n+1)² is 2n. Here, n=12, so there are 2 * 12 = 24 numbers.
• Find the square root of 900 by prime factorization. 900 = 9 x 100 = 3 x 3 x 10 x 10 = (3 x 10) x (3 x 10) = 30 x 30. So, √900 = 30.
• What is the smallest number that must be subtracted from 1989 to make it a perfect square? Find √1989 by long division. 44² = 1936 and 45² = 2025. 1989 is closer to 1936. Remainder = 1989 - 1936 = 53. So, 53 must be subtracted.

Frequently Asked Questions

Frequently Asked Questions