CLASS-VII CHAPTER-4 Exponents and powers Part - 1 An exponent or power is a mathematical representation that indicates the number of times that a number is multiplied by itself. An exponent or power is a mathematical representation that indicates the number of times that a number is multiplied by itself . If a number is multiplied by itself m times, then it can be written as: a x a x a x a x a...m times = a m Here, a is called the base , and m is called the exponent, power or index. Numbers raised to the power of two are called square numbers. Square numbers are also read as two-square, three-square, four-square, five-square, and so on. Numbers raised to the power of three are called cube numbers. Cube numbers are also read as two-cube, three-cube, four-cube, five-cube, and so on. Negative numbers can also be written using exponents. If a n = b, where a and b are integers and n is a natural number , then a n is called the exponential form of b. The factors of a product can be expressed as the powers of the prime factors . This form of expressing numbers using exponents is called the prime factor product form of exponents. Even if we interchange the order of the factors , the value remains the same. So a raised to the power of x multiplied by b raised to the power of y, is the same as b raised to the power of y multiplied by a raised to the power of x. The value of an exponential number with a negative base raised to the power of an even number is positive. If the base of two exponential numbers is the same, then the number with the greater exponent is greater than the number with the smaller exponent. EXERCISE 4.1 Question 2. Find the value of the following: (i) 2 6 (ii) 5 5 (iii) (-6) 4 Solution: Question 3. Express the following in the exponential form: (iii) 2 × 2 × a × a × a × a (iv) a × a × a × c × c × c × c × d Solution: Question 4. Simplify the following: (iii) 3 3 × 10 4 Solution: Question 5. Simplify the following: (ii) (-3) 2 × (-5) 2 (iv) (-1) 9 (vi) 4 2 × 3 3 × (-1) 122 Solution: Question 6. Identify the greater number in each of the following: (iii) 4 5 or 5 4 (iv) 2 10 or 10 2 Solution: Question 7. Write the following numbers as powers of 2: (iii) 1024 Solution: Question 9. Write the following numbers as powers of (-3): (iii) 81 Solution: Question 10. Find the value of x in each of the following: (ii) 3 x = 729 (iv) (-4) x = -1024 (vi) (−34 )x = −2431024 Solution: Question 11. Write the prime factorization of the following numbers in the exponential form: (ii) 360 (iv) 540 (vi) 3600 (viii) 8400 Solution: PRACTICE QUESTIONS Question 1. Fill in the blanks: (i) In the expression 3 7, base = .. and exponent = .. (ii) In the expression (-7) 5, base = . and exponent = . (iii) In the expression (25 )11 , base = and exponent = . (iv) If base is 6 and exponent is 8, then exponential form = .. Question 2. Find the value of the following: (vi) (23 )4 (v) (−23 )5 (vi) (-2) 9 Question 3. Express the following in the exponential form: (i) 6 × 6 × 6 × 6 × 6 (ii) t × t × t Question 4. Simplify the following: (i) 7 × 10 3 (ii) 25 × 9 Question 5. Simplify the following: (i) (-3) × (-2) 3 (iii) (-2) 3 × (-10) 4 (v) 25 2 × (-1) 31 Question 6. Identify the greater number in each of the following: (i) 4 3 or 3 4 (ii) 7 3 or 3 7 Question 7. Write the following numbers as powers of 2: (i) 8 (ii) 128 Question 8. To what power (-2) should be raised to get 16? Question 9. Write the following numbers as powers of (-3): (i) 9 (ii) -27 Question 10. Find the value of x in each of the following: (i) 7 x = 343 (i ii ) (-8) x = -512 (v) (25 )x = 323125 Question 11. Write the prime factorization of the following numbers in the exponential form: (i) 72 (iii) 405 (v) 2280 (vii) 4725 .