UNIT 3 SQUARE-SQUARE ROOT AND CUBE-CUBE ROOT (A) Main Concepts and Results • A natural number is called a perfect square if it is the square of some natural number. i.e., if m = n2, then m is a perfect square where m and n are natural numbers. • A natural number is called a perfect cube if it is the cube of some natural number. i.e., if m = n3, then m is a perfect cube where m and n are natural numbers. • Number obtained when a number is multiplied by itself is called the square of the number. • Number obtained when a number is multiplied by itself three times are called cube number. • Squares and cubes of even numbers are even. • Squares and cubes of odd numbers are odd. • A perfect square can always be expressed as the product of pairs of prime factors. • A perfect cube can always be expressed as the product of triplets of prime factors.
MATHEMATICS • The unit digit of a perfect square can be only 0, 1, 4, 5, 6 or 9. • The square of a number having: 1 or 9 at the units place ends in1. 2 or 8 at the units place ends in 4. 3 or 7 at the units place ends in 9. 4 or 6 at the units place ends in 6. 5 at the units place ends in 5. • There are 2n natural numbers between the squares of numbers n and n+1. • A number ending in odd numbers of zeroes is not a perfect square. • The sum of first n odd natural numbers is given by n2. • Three natural numbers a, b, c are said to form a pythagorean triplet if a2 + b2 = c2. • For every natural number m > 1, 2m, m2–1 and m2 + 1 form a pythagorean triplet. • The square root of a number x is the number whose square is x. Positive square root of a number x is denoted by x. • The cube root of a number x is the number whose cube is x. It is denoted by 3 x. • Square root and cube root are the inverse operations of squares and cubes respectively. • If a perfect square is of n digits, then its square root will have n 2 ⎛ n +1⎞ digit if n is even or ⎜ ⎟ digit if n is odd. ⎝ 2 ⎠ • Cubes of the numbers ending with the digits 0, 1, 4, 5, 6 and 9 end with digits 0, 1, 4, 5, 6 and 9 respectively. Think 74 and Discuss 1. Describe what is meant by a perfect square. Give an example. 2. Explain how many square roots a positive number can have. How are these square roots different? EXEMPLAR PROBLEMS
UNIT-3 Key Concept To be Noted SQUARE ROOTS Words A square root of a number n is a number m which, when multiplied by itself, equals n. Numbers The square roots of 16 are 4 and – 4 because 42 = 16 and (– 4)2 = 16. Algebra If m2 = n, then m is a square root of n. Think and Discuss 1. Which type of number has an exact square root? 2. Which type of number has an approximate square root? 3. How can we use perfect squares to estimate a square root, such as 8 ? • Cube of the number ending in 2 ends in 8 and cube root of the number ending in 8 ends in 2. • Cube of the number ending in 3 ends in 7 and cube root of the number ending in 7 ends in 3. (B) Solved Examples In examples 1 to 7, out of given four choices only one is correct. Write the correct answer. Example 1 : Which of the following is the square of an odd number? (a) 256 Solution (b) 361 (c) 144 (d) 400 : Correct answer is (b). Example 2 : Which of the following will have 1 at its units place? (a) 192 Solution (b) 172 (c) 182 (d) 162 : Correct answer is (a). Example 3 : How many natural numbers lie between 182 and 192? (a) 30 Solution (b) 37 (c) 35 (d) 36 : Correct answer is (d). SQUARE-SQUARE ROOT AND CUBE-CUBE ROOT 75
MATHEMATICS Example 4 : Which of the following is not a perfect square? (a) 361 Solution (b) 1156 (c) 1128 (d) 1681 : Correct answer is (c). Example 5 : A perfect square can never have the following digit at ones place. (a) 1 Solution (b) 6 (c) 5 (d) 3 : Correct answer is (d). Example 6 : The value of 176 + 2401 is (a) 14 Solution (b) 15 (c) 16 : Correct answer is (b). ( 176 + 2401 = 176 + 49 = 225 = 15 Example 7 : Given that 5625 =75, the value of (a) 82.5 Solution (d) 17 (b) 0.75 ) 0.5625 + 56.25 is: (c) 8.25 (d) 75.05 : Correct answer is (c). If ( 5625 = 75, then 0.5625 = 0.75 and 56.25 = 7.5) In examples 8 to 14, fill in the blanks to make the statements true. Example 8 : There are __________ perfect squares between 1 and 50. Solution : 6 Example 9 : The cube of 100 will have __________ zeroes. Solution : 6 Example 10 : The square of 6.1 is ____________. Solution 76 : 37.21 1. Squaring a number and taking a square root are inverse operations. What other inverse operations do you know? 2. When the factors of a perfect square are written in order from the least to greatest, what do you notice? 3. Why do you think numbers such as 4, 9, 16, ... are called perfect squares? 4. Suppose you list the factors of a perfect square. Why is one factor square root and not the other factors? EXEMPLAR PROBLEMS