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Unit 11(Exponents & Powers)

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Unit 11(Exponents & Powers) • Exponents are used to express large numbers in shorter form to make them easy to read, understand, compare and operate upon. • a × a × a × a = a4 (read as ‘a’ raised to the exponent 4 or the fourth power of a), where ‘a’ is the base and 4 is the exponent and a4 is called the exponential form. a × a × a × a is called the expanded form. • For any non-zero integers ‘a’ and ‘b’ and whole numbers m and n, (i) am × an = am+n (ii) am ÷ an = am–n , m>n (iii) (am)n = amn (iv) am × bm = ( ab)m m a (v) am ÷ bm = b (vi) a0 = 1 (vii) (–1)even number = 1 (viii) (–1)odd number = –1 • Any number can be expressed as a decimal number between 1.0 and 10.0 (including 1.0) multiplied by a power of 10. Such form of a number is called its standard form or scientific notation. 15-04-2018 In Examples 1 to 3, there are four options, out of which one is correct. Write the correct one. – 8 Example 1: Out of the following, the number which is not equal to is 27 3 3 2 −2 (a) – (b) 3 3 3 −2 −−−222 (c) – (d) ×× 3 333 Solution: Correct answer is (c). Example 2: ()()−775 ×− 3 is equal to (a) ()−7 8 (b) – ()7 8 (c) ()−7 15 (d) ()−7 2 Solution: Correct answer is (a). Example 3: For any two non-zero integers x any y, x3 ÷ y3 is equal to 0 3 æöx x (a) ç÷ (b) èøy y 6 9 æöx æöx (c) èøç÷y (d) èøç÷y Solution: Correct answer is (b). Words Numbers Algebra To multiply powers m n m+n with the same base, 35 × 38 = 3 5 + 8 = 3 13 b × b = b keep the base and add the exponents. In Examples 4 and 5, fill in the blanks to make the statements true. 2 Example 4: (5576÷ ) = ________ Solution: 52 15-04-2018 ab73 Example 5: = __________ ab5 Solution: (ab)2 In Examples 6 to 8, state whether the statements are True or False: Example 6: In the number 75, 5 is the base and 7 is the exponent. Solution: False A power is written in base and exponent form as follows: The base is the number that Þ The exponent tells you is being repeated as a factor Þ 52 how many times the base in the multiplication. is repeated as a factor in the multiplication For example 7.7 = 72, 3•3•3•3•3 = 35. a4 aaaa+++ Example 7: = b3 bbb++ Solution: False Example 8: ab > b a is true, if a = 3 and b = 4; but false, if a = 2 and b = 3. Solution: True Example 9: By what number should we multiply 3 3 so that the product may be equal to 37? Solution: Let 33 be multiplied by x so that the product may be equal to 37. According to question, 3 3 × x = 37 or x = 37 ÷ 33 = (3)7–3 (Using am ÷ an = (a)m–n ) = 34 = 81 Therefore, 33 should be multiplied by 81 so that the product is equal to 37. 15-04-2018 5 11 8x 55 5 Example 10: Find x so that ×= 33 3 5 11 8x 55 5 Solution: Given ×= 33 3 m 555 11 58 x aam ×= Using = So, 5 11 m 33 3 b b 555× 11 58 x = or 5 11 33× 3 8 x (5)16 5 m n m+n or 16 = {Using a × a = (a) } (3)3 16 8 x 55 or = 33 or 16 = 8x Thus, 8x = 16 Therefore, x = 2 Example 11: Express 648 in exponential notation. 2 648 Solution: 648 = 2 × 2 × 2 × 3 × 3 × 3 × 3 2 324 = 23 × 34 2 162 Express 2,36,00,000 in standard Example 12: 3 81 form. Solution: 236,00,000 3 27 236,00,000 39 = ×100,00,000 100,00,000 33 = 2.36 × 107 1 Example 13: Which of the two is larger : 312 or 66 ? Solution: 312 = 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 = 531441 66 = 6 × 6 × 6 × 6 × 6 × 6 = 46656 So, 312 is greater. 15-04-2018 Example 14 1 5 1 19 1 8x Find x such that × = 5 5 5 Solution: Understand and Explore the Problem • What are you trying to find? The value of x for the given equation. Plan a Strategy • You know the laws of exponents. Apply those laws in the given equation to find the value of x. Solve 1 5 1 19 1 8x • Given, × = 5 5 5 Using the law of exponents, am × an = am+n, we get + 1 5 19 1 8x = 5 5 1 24 1 8x = 5 5 On both the sides, powers have the same base. So, their exponents must be equal Therefore, 24 = 8x 24 or x==3 8 Hence, the value of x is 3. 15-04-2018 Revise • Substitute the value of x in the equation and check if it satisfies the equation. + 1 5 1 19 1 5 19 1 24 LHS = × = = 5 5 5 5 × 1 8x 1 83 1 24 RHS = = = 5 5 5 LHS = RHS Hence, the equation is satisfied with x = 3. So, our answer is correct. 1 3 1. Try to find the value of x given in the question by changing to . 5 2 What difference do you find in the value of x ? What do you infer from your answer? 2. Can you find the value of x if the equation is changed to (5)x ÷ (5)2 = (5)3 ? In questions 1 to 22, there are four options, out of which one is correct. Write the correct one. 1. [(–3)2]3 is equal to (a) (–3)8 (b) (–3)6 (c) (–3)5 (d) (–3)23 2. For a non-zero rational number x, xx82÷ is equal to (a) x4 (b) x6 (c) x10 (d) x16 3. x is a non-zero rational number. Product of the square of x with the cube of x is equal to the (a) second power of x (b) third power of x (c) fifth power of x (d) sixth power of x 15-04-2018 4. For any two non-zero rational numbers x and y, xy55÷ is equal to (a) (x÷y)1 (b) (x÷y)0 (c) (x÷y)5 (d) (x÷y)10 5. am × an is equal to (a) (a2)mn (b) am–n (c) am+n (d) amn 6. (10 + 20 + 30) is equal to (a) 0 (b) 1 (c) 3 (d) 6 1022+ 10 20 7. Value of is 1020 (a) 10 (b) 1042 (c) 101 (d) 1022 8. The standard form of the number 12345 is (a) 1234.5 × 101 (b) 123.45 × 102 (c) 12.345 × 103 (d) 1.2345 × 104 9. If 21998 – 21997 – 21996 + 21995 = K.21995, then the value of K is (a) 1 (b) 2 (c) 3 (d) 4 10. Which of the follwing is equal to 1? (a) 20 + 30 + 40 (b) 20 × 30 × 40 (c) (30 – 2 0) × 40 (d) (30 – 2 0) × (30 +20) 11. In standard form, the number 72105.4 is written as 7.21054 × 10n where n is equal to (a) 2 (b) 3 (c) 4 (d) 5 −2 12. Square of is 3 −2 2 −4 4 (a) (b) (c) (d) 3 3 9 9 Words Numbers Algebra To divide powers with the same base, keep m 9 b mn– 6 9–4 5 =b the base and subtract ==66 n 4 b the exponents. 6 15-04-2018 Product of Powers Property Words To multiply powers with the same base, add their exponents. Algebra am . an = am + n Numbers 56 . 53 = 56 + 3 = 59 −1 13. Cube of is 4 –1 1 −1 1 (a) (b) (c) (d) 12 16 64 64 4 – 5 14. Which of the following is not equal to ? 4 4 (– 5)4 5 (a) (b) 4 44 ( – 4 ) 4 5 55 5 5 (c) – (d) − ×− × − ×− 44 4 4 4 4 15. Which of the following is not equal to 1 ? 32 23× 7 −34 ×− ÷−()2 (a) 4× 18 (b) ()()22 303 ×5 24 (c) (d) 0 03 5×25 (7 +3 ) 33 25 16. × is equal to 37 9 6 3 0 25 25 25 25 (a) × (b) × (c) × (d) × 37 37 37 37 17. In standard form, the number 829030000 is written as K × 108 where K is equal to (a) 82903 (b) 829.03 (c) 82.903 (d) 8.2903 15-04-2018 18. Which of the following has the largest value? 1 1 1 (a) 0.0001 (b) (c) (d) ÷ 0.1 10000 106 106 19. In standard form 72 crore is written as (a) 72 × 107 (b) 72 × 108 (c) 7.2 × 108 (d) 7.2 × 107 mn aa 20. For non-zero numbers a and b, ÷ , where m > n, is equal to bb mn m + n m – n m n a a a a (a) (b) (c) (d) b b b b 21. Which of the following is not true? (a) 32 > 23 (b) 43 = 26 (c) 33 = 9 (d) 25 > 52 22. Which power of 8 is equal to 26? (a) 3 (b) 2 (c) 1 (d) 4 In questions 23 to 39, fill in the blanks to make the statements true. 23. (–2)31 × (–2)13 = (–2) 24. (–3)8 ÷ (–3)5 = (–3) 4 9 3 _____ 11 11 5 11 –1 –1 –1 25.
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