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

Info icon This preview shows pages 10–12. Sign up to view the full content.

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
Image of page 10

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

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
Image of page 11
Image of page 12
This is the end of the preview. Sign up to access the rest of the document.
  • Winter '10
  • John Schaefer
  • Math, Open Learning, Open Learning Unit, Learning Unit Level

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern