# B using the third law of indices we can write 32 2 5

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b) Using the third law of indices we can write 32 2 / 5 = 32 2 × 1 5 = (32 1 5 ) 2 . Thus 32 2 / 5 = ((32) 1 / 5 ) 2 = 2 2 = 4 c) Note that 8 1 / 3 = 2. Then 8 2 3 = 8 2 × 1 3 = (8 1 / 3 ) 2 = 2 2 = 4 Note the following alternatives: 8 2 / 3 = (8 1 / 3 ) 2 = (8 2 ) 1 / 3 Example Write the following as a simple power with a single index: a) x 5 , b) 4 x 3 . Solution a) x 5 = ( x 5 ) 1 2 . Then using the third law of indices we can write this as x 5 × 1 2 = x 5 2 . b) 4 x 3 = ( x 3 ) 1 4 . Using the third law we can write this as x 3 × 1 4 = x 3 4 . Example Show that z - 1 / 2 = 1 z . Solution z - 1 / 2 = 1 z 1 / 2 = 1 z Try each part of this exercise Simplify z z 3 z - 1 / 2 Part (a) Rewrite z using an index and simplify the denominator using the first law of indices: Answer Part (b) Finally, use the second law to simplify the result: Answer Engineering Mathematics: Open Learning Unit Level 0 1.2: Basic Algebra 10

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Example The generalisation of the third law of indices states that ( a m b n ) k = a mk b nk . By taking m = 1, n = 1 and k = 1 2 show that ab = a b . Solution Taking m = 1, n = 1 and k = 1 2 gives ( ab ) 1 / 2 = a 1 / 2 b 1 / 2 and the required result follows immediately. Key Point ab = a b This result often allows answers to be written in alternative forms. For example we may write 48 as 3 × 16 = 3 16 = 4 3. Although this rule works for multiplication we should be aware that it does not work for addition or subtraction so that a ± b 6 = a ± b More exercises for you to try 1. Evaluate using a calculator a) 3 1 / 2 , b) 15 - 1 3 , c) 85 3 , d) 81 1 / 4 2. Evaluate using a calculator a) 15 - 5 , b) 15 - 2 / 7 3. Simplify a) a 11 a 3 / 4 a - 1 / 2 , b) z z 3 / 2 , c) z - 5 / 2 z , d) 3 a 2 a , e) 5 z z 1 / 2 . 4. From the third law of indices show that ( ab ) 1 / 2 = a 1 / 2 b 1 / 2 . Deduce that the square root of a product is equal to the product of the individual square roots. 5. Write each of the following expressions with a single index: a) ( x - 4 ) 3 , b) x 1 / 2 x 1 / 4 , c) x 1 / 2 x 1 / 4 Answer 6. Scientific notation It is often necessary to use very large or very small numbers such as 78000000 and 0.00000034. Scientific notation can be used to express such numbers in a more concise form. Each number is written in the form a × 10 n where a is a number between 1 and 10. We can make use of the following facts: 10 = 10 1 , 100 = 10 2 , 1000 = 10 3 and so on and 0 . 1 = 10 - 1 , 0 . 01 = 10 - 2 , 0 . 001 = 10 - 3 and so on Furthermore, to multiply a number by 10 n the decimal point is moved n places to the right if n is a positive integer, and n
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• Winter '10
• John Schaefer
• Math, Open Learning, Open Learning Unit, Learning Unit Level

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