Solution a 2 3 1 2 3 1 8 b 1 4 3 4 3 64 c x 1 1 x 1 1

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Solution a) 2 - 3 = 1 2 3 = 1 8 , b) 1 4 - 3 = 4 3 = 64, c) x - 1 = 1 x 1 = 1 x , d) x - 2 = 1 x 2 , e) 10 - 1 = 1 10 1 = 1 10 or 0.1. Now do this exercise Write each of the following using a positive index. a) 1 t - 4 , b) 17 - 3 , c) y - 1 , d) 10 - 2 Use the previous Key Point. Answer Try each part of this exercise Simplify a) a 8 × a 7 a 4 , b) m 9 × m - 2 m - 3 Part (a)(i) Use the first law of indices to simplify the numerator: Answer Part (a)(ii) Then use the second law to simplify the result: Answer Part (b)(i) First simplify the numerator using the first law of indices: Answer Part (b)(ii) Then use the second law to simplify the result: Answer Engineering Mathematics: Open Learning Unit Level 0 1.2: Basic Algebra 8
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More exercises for you to try 1. Write the following numbers using a positive index and also express your answers as decimal fractions: a) 10 - 1 , b) 10 - 3 , c) 10 - 4 2. Simplify as much as possible: a) x 3 x - 2 , b) t 4 t - 3 , c) y - 2 y - 6 . Answer 5. Fractional indices. So far we have used indices that are whole numbers. We now consider fractional powers. Con- sider the expression (16 1 2 ) 2 . Using the third law of indices, ( a m ) n = a mn , we can write (16 1 2 ) 2 = 16 1 2 × 2 = 16 1 = 16 So 16 1 2 is a number which when squared equals 16, that is 4 or - 4. In other words 16 1 2 is a square root of 16. There are always two square roots of a non-zero positive number, and we write 16 1 2 = ± 4 In general Key Point a 1 2 is a square root of a Similarly (8 1 3 ) 3 = 8 1 3 × 3 = 8 1 = 8 so that 8 1 3 is a number which when cubed equals 8. Thus 8 1 3 is the cube root of 8, that is 3 8, namely 2. Each number has only one cube root, and so 8 1 3 = 2 In general Key Point a 1 3 is the cube root of a More generally we have Key Point x 1 n is an n th root of x, that is n x Your calculator will be able to evaluate fractional powers, and roots of numbers. Check that you can obtain the results of the following examples on your calculator, but be aware that your calculator may give only one root when there may be others. 9 Engineering Mathematics: Open Learning Unit Level 0 1.2: Basic Algebra
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Example Evaluate a) 144 1 / 2 , b) 125 1 / 3 Solution a) 144 1 / 2 is a square root of 144, that is ± 12 . b) Noting that 5 3 = 125, we see that 125 1 / 3 = 3 125 = 5 Example Evaluate a) 32 1 / 5 , b) 32 2 / 5 , and c) 8 2 / 3 . Solution a) 32 1 5 is the 5th root of 32, that is 5 32. Now 2 5 = 32 and so 5 32 = 2.
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