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Indices

Indices. Chapter 1 2014 – Year 10 Mathematical Methods. Review of Index Laws. Some numbers can be written in mathematical shorthand if the number is the product of "repeating numbers”. Example : a 7 = a × a × a × a × a × a × a = aaaaaaa Index and base form 64 = 2 6

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Indices

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  1. Indices Chapter 1 2014 – Year 10 Mathematical Methods

  2. Review of Index Laws Some numbers can be written in mathematical shorthand if the number is the product of "repeating numbers”. Example: a7= a × a × a × a × a × a × a = aaaaaaa Index and base form 64 = 26 • The 10 is called the index number • The 2 is called the base number The plural of “index” is “indices” Another name for index form is power form or power notation 26 is read as: two to the power of 6

  3. Review of Index Laws Index Law 2 26 24 In general terms am an Index Law 1 23 x 25 = 23+5 = 28 In general terms am x an = am+n Index Law 3 23 23 In general terms a0 = 1 = 23-3 =20 = 1 = 26-4 =22 = am-n

  4. Review of Index Laws Index Law 4 (24)2 = 24 X 2 = 28 In general terms (am)n = am x n = amn Index Law 6 In general terms Index Law 5 (2 x 3)4 = 23 x 34 In general terms (a x b)m = am x bm

  5. Examples Solve: m2n6p2 x m3np4 = m2+3n6+1p2+4 = m5n7p6 Solve: 6x3y5 2xy2 =3x3-1y5-2 =3x2y3

  6. Examples Which of the following is equivalent to (x½)6? A. x6½ B. x3 C.6x½ D. ½x6 We get: = x½x 6 = x3 B Using law 4 (am)n = am x n = amn

  7. Examples Which of the following is equivalent to (2y⅔)3? A. 8y2 B. 2y2 C.8y3 D. 2y3 Using law 4 (am)n = am x n = amn We get: = 23y⅔ x 3 = 8y2 A

  8. Negative Indices Lets have a look at this example of Index Law 2 y2y x y1 y4 y x y x y x yy2 Therefore we know y-2 also can be written as Seventh Index Law a-n = It can also be written as =y-2 or 1 y2 1 an

  9. Negative Indices • All index laws apply to terms with negative indices • Always express answers with positive indices unless otherwise instructed • Numbers and pronumerals without an index are understood to have an index of 1 e.g. 2 = 21

  10. Examples Write the numerical value of: Express the following with a positive index:

  11. Examples • Simplify these algebraic expression: HINT – remove the brackets first, then use the index laws and then express with positive indices.

  12. Fractional Indices • Fractional indicesarethosewhichareexpressedasfractions.

  13. Fractional Indices

  14. Fractional Indices

  15. Fractional Indices

  16. Combining Index Laws When more than one index law is used to simplify an expression, the following steps can be taken. Step 1: If an expression contains brackets, expand them first. Step 2: If an expression is a fraction, simplify each numerator and denominator, then divide (simplify across then down). Step 3: Express the final answer with positive indices.

  17. Combining Index Laws Simplify :

  18. Combining Index Laws Simplify:

  19. Combining Index Laws Simplify:

  20. Combining Index Laws Simplify:

  21. Combining Index Laws Simplify:

  22. Combining Index Laws Simplify:

  23. Combining Index Laws • Simplification of expressionswithindicesofteninvolvesapplicationofmorethanoneIndex law. • If anexpressioncontainsbrackets, theyshouldberemovedfirst. • If theexpressioncontainsfractions, simplifyacrossthendown. • Whendividingfractions, change÷ to × and flipthesecondfraction(multiply and flip). • Expressthefinal answerwithpositive indices.

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