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Chapter Analysis
Intermediate21 pages • EnglishQuick Summary
Chapter 5 introduces the concepts of squares and square roots. It explains how squares of numbers are formed by multiplying the number by itself, and details methods such as prime factorization and long division for finding square roots. The chapter also covers perfect squares and provides exercises to calculate the square roots of various numbers. Practical applications include estimating square roots and identifying numbers that are not perfect squares.
Key Topics
- •Understanding perfect squares
- •Methods of finding square roots
- •Prime factorization
- •Estimating square roots
- •Properties of square numbers
- •Computation of square roots using long division
- •Pythagorean triplets
- •Patterns in square numbers
Learning Objectives
- ✓Identify and calculate perfect squares and square roots.
- ✓Apply various methods to find square roots, including prime factorization and division.
- ✓Understand the properties and patterns of square numbers.
- ✓Determine whether a number is a perfect square.
- ✓Solve practical problems involving squares and square roots.
- ✓Derive and work with Pythagorean triplets.
Questions in Chapter
What will be the unit digit of the squares of the following numbers? (i) 81 (ii) 272 (iii) 799 (iv) 3853 (v) 1234 (vi) 26387 (vii) 52698 (viii) 99880 (ix) 12796 (x) 55555
Page 52
The following numbers are obviously not perfect squares. Give reason. (i) 1057 (ii) 23453 (iii) 7928 (iv) 222222 (v) 64000 (vi) 89722 (vii) 222000 (viii) 505050
Page 52
Find the square roots of 100 and 169 by the method of repeated subtraction.
Page 54
Find the least number which must be added to each of the following numbers so as to get a perfect square. (i) 525 (ii) 1750 (iii) 252 (iv) 1825 (v) 6412
Page 69
In a right triangle ABC, ∠B = 90°. If AB = 6 cm, BC = 8 cm, find AC.
Page 70
Additional Practice Questions
Explain the method of finding the square root of a number using prime factorization.
mediumAnswer: To find the square root using prime factorization, factorize the given number into prime numbers. Pair these factors and take one from each pair to multiply together. The product is the square root of the given number.
How can you determine if a number is a perfect square by looking at its units digit?
easyAnswer: A number is a perfect square if its unit digit ends in 0, 1, 4, 5, 6, or 9. For instance, 64, 81, and 100 are perfect squares because they end in 4, 1, and 0, respectively.
What is the smallest perfect square that is divisible by each of the numbers 6, 9, and 15?
mediumAnswer: First, find the LCM of 6, 9, and 15, which is 90. Then multiply 90 by the smallest number needed to make it a perfect square, which is 10. Thus, the smallest perfect square is 900.
Find a Pythagorean triplet where the smallest member is 8.
hardAnswer: Using the formula 2m, m²-1, and m²+1, if 2m = 8, then m = 4, giving the triplet 8, 15, 17.
Without calculation, explain how to estimate the number of digits in the square root of a perfect square.
easyAnswer: To estimate the number of digits, place bars over the digits in pairs from right to left. The number of bars determines the number of digits in the square root.