Basic Concepts of Algebra

4. INDICES II - More on Exponents

4.    INDICES II – More on Exponents.

4.1  THE LAWS OF INDICES:
        In algebra the words indices, exponents and powers are used interchangeably. In this chapter, for further work on chapter 3, we will use the term indices instead. There are six basic laws of Indices. We will use examples to explain these laws and learn how to simplify expressions with indices.
      Laws of Indices:
1.  a^m∙a^n  =a^(m+n)
2.  (a^m )^n  =a^mn
3.  a^m÷a^n=a^(m-n)
4.    a^0=1,where a≠o
5.  a^(-n)=1/a^n
6.  a^ m/n= (〖nth√(a))〗^m or nth√(a^m )





1. Law # 1. a^m∙a^n=a^(m+n)
    Consider: a^3∙a^4
                      a^3=a a a and a^4=a a a a
                      a^3∙a^4=a a a  a a a a=a^7
    The index 7 in the answer is the sum 3 + 4 of the indices.
    Hence: a^3∙a^4=a^(3+4)=a^7
    Examples:
1. a^2∙a^3=a a a a a=a^5 or a^(2+3)=  a^5 Law #1
2. To simplify 3a^2 x 5a^3, consider first what it means.
    3a^2 x 5a^3→(3xa^2 )x(5xa^3 )→(3x5)x(a^2 xa^3 )
                                                      → 15 a^(2+3)
                                                      →15a^5
2. Law #2. (a^m )^n=a^mn
    Consider: (a^3 )^4
                    The index 4 tells us how many times a^3 is to be taken as a factor.
    Thus, (a^3 )^4=aaa aaa aaa aaa=a^12
                          =(a^3 )(a^3 )(a^3 )(a^3 )
    Note that the index 12 is the product of the indices 3 by 4.
    Hence: (a^3 )^4=a^3x4=a^12  Law #2
    Examples:
1. (a^5 )^2=aaaaa  aaaaa=a^10
                =(a^5 )(a^5 )
    Thus, a^5x2 =a^10
2. (a^15 )^2 =a^15x2=a^30
    (a^15 )^2=(a^15 )(a^15 )=a^(15+15)=a^30
    Extension of Law 2
1.  Power of a product (ab)^n=a^n b^n
    Consider: (4x^3 )^2                        The index 2 tells us how many times 4x^3 is to be taken as a factor.
    Thus, (4x^3 )^2=(4x^3 )(4x^3 )=(4 x 4)(x^3 x x^3 )
                                                        =16x^(3+3)
                                                        =16x^6
    Alternatively:
    (4x^3 )^2=4^1x2  x^3x2=4^2 x^6=16x^6
    Example:
    Simplify: (3x^2 y^3 )^4
    (3x^2 y^3 )^4=(3x^2 y^3 )(3x^2 y^3 )(3x^2 y^3 )(3x^2 y^3 )
                        =3 x 3 x 3 x 3 x^2  x x^2  x x^2  x x^2 x y^3  x y^3  x y^3 x y^3
                        =81x^8 y^12
   
    Alternatively:
    (3x^2 y^3 )^4=3^1x4  x^2x4  y^3x4=3^4 x^8 y^12=81x^8 y^12
2.  Power of a fraction (a/b)^n=a^n/b^n
    Consider: (x^2/y^3 )^4
                      The index 4 tells us how many times x^2/y^3  is to be taken as a factor.

      Thus, (x^2/y^3 )^4=(x^2/y^3 )(x^2/y^3 )(x^2/y^3 )(x^2/y^3 )
              =(x^2  x^2  x^2  x^2)/(y^3  y^3  y^3  y^3 )=x^8/y^12
   
      Alternatively:
                              (x^2/y^3 )^4=x^2x4/y^3x4 =x^8/y^12
      Example:
                              ((5t^3)/(7w^2 ))^2=((5t^3)/(7w^2 ))((5t^3)/(7w^2 ))
                                            =(5 x 5 x t^3 x t^3)/(7 x 7 x w^2  x w^2 )=(25t^6)/(49w^4 )
     
    Alternatively:
                            ((5t^3)/(7w^2 ))^2=(5^1x2 t^3x2)/(7^1x2 w^2x2 )=(5^2 t^6)/(7^2 w^4 )=(25t^6)/(49w^4 )           

3. Law #3. a^m/a^n =a^(m-n)  or 1/a^(n-m) 
    Consider: a^5/a^3 =aaaaa/aaa=a^2
    Simply a^5/a^3 =a^(5-3)=a^2  (shortest form)
   
    Examples:
1.  x^12/x^7 = x^(12-7)=x^5

2.  x^9/x^13 =x^(9-13)=x^(-4)=1/x^4
   
 
  Alternatively:
                            x^9/x^13 =1/x^(13-9 ) =1/x^4  since a^m/a^n =a^(1/n)-m

4. Law #4. a^0=1,where a≠o
    Consider:          7×2 =14                      opp. of multiplication is division
                             
                              14÷2=7
                      a^0  ×  a^3 =〖 a〗^3
                      a^3 ÷ a^3 = 1

    Examples:
1. x^0=1
2. 5^0=1
3. (x^3 )^0=1
4. (5x^2 y)^0=1

5. Law #5. a^(-n)=1/a^n
    Consider: a^(-3) x  a^3=a^0=1
                      1÷a^3=1/a^3
Therefore, a^(-3)=1/a^3
Hence,      a^(-n)=1/a^n  and a^n=1/a^(-n)
    Examples:
1. 2^(-1)=1/2
2. t^(-3)=1/t^3
3. 2ab^(-2)=2a x 1/b^2 =2a/b^2
4. 1/a^(-2) =a^2
5. 7/(x^(-3) y)=(7x^3)/y

6. Law #6. a^(m/n)=(√(n&a))^m  or √(n&a^m )
    Consider: 8^(2/3)
                      The number 3 (the root) tells us to take the cube root of the base 8 and the number 2 (the index) tell us to square the result. We may otherwise take the cube root of the base squared. Thus, 8^(2/3)=(∛8)^2  or ∛(8^2 )
                                        =2^2
                                        =4
    Examples:
1. 〖27〗^(2/3)=(∛27)^2=3^2=9
2. 〖64〗^(2/3)=(∛64)^2=4^2=16
3. 〖9^-〗^(1/2)=(√9)^(-1)=3^(-1)=1/3  or, alternatively, =1/9^(1/2) =1/√9=1/3
4. 5^(2/3)=(∛5)^2=∛(5^2 )=∛25 (radical form)

   


    Exercise 4.1
    Simplify the following:

1. a^5∙a^2

2. a^5∙a^(-2)

3. 3a^5  x 3a^2

4. 5a^3  x 2a^4

5. (a^3 )^5

6. (a^5 )^3

7. (5x^3 )^2

8. (2x^3 y^2 )^4

9. (4p^2 y^3 )^2

10. (a^2/b^3 )^5

11. (〖5s〗^3/(7t^2 ))^2

12. a^5÷a^2

13. b^5/b^2

14. x^9/x^13

15. (5x^2 y)^0

16. 2ab^(-2)

17. 9/(x^(-3) y)

18. 〖125〗^(2/3)

19. 〖64〗^(5/6)

20. 〖49〗^(-1/2)
   




4.2 SIMPLIYING EXPRESSIONS WITH FRACTIONAL INDICIES:
      To simplify, expressions involving indices means to write the final answer without using                                             
      zero, negative, or fractional indices.
      Examples:
1.    Simplify: 〖(x^(-1/2) )^-〗^(2/3)
      (x^(-1/2) )^(-2/3)=x^(-1/2 x-2/3 )=x^(1/3)=∛x 
       
      Remove parentheses; multiply indices; rewrite as a radical
2.    Simplify: (9x^(-4) )^(1/2)
      (9x^(-4) )^(1/2)=(9^(1/2) )(x^(-4x 1/2  ) )=9^(1/2)∙x^(-2)=9^(1/2)/x^2 =3/x^2

      Remove parentheses; multiply indices; rewrite with positive indices.
3.    Simplify: (x^6 y^(-3) )^(-2/3)
      (x^6 y^(-3) )^(-2/3)=x^(6x-2/3) y^(-3x-2/3)=x^(-4) y^2=y^2/x^4
      Remove parentheses; multiply indices; rewrite with positive indices.
     




      EXERCISE 4.2
      Simplify the following:
1.  (∛7)^6

2.    ∛(x^2 )/∜x

3.    (x^(1/3) )^3

4.    (5^°/8^(1/3) )^(-1)

5.[  (〖〖√2/(-3))〗^(-4)]〗^(-1)

6.    (√x/x^2 )^(-2)

7.    (y^(3/5) )^(1/4)


8.    x^(1/2)∙x^(5/2)

9.  (8√x)^(-2/3) 

10.  (x^(1/3)/x^(2/3) )^3




4.3    SUMMARY EXCERISE FOR Ch 4.
          Simply the following where possible:

1.  x^3  × x^4                                                      20.  (a^(-9) b^(-8))/(a^(-4) b^3 )

2.  x^4  × x^6                                                      21.  (3xy^(-2) )^(-3)/x

3.  x^3  × x                                                          22.  [(ab)^(-2)/(a^- 1)]^3

4.  (2w^3 x^5 )(5x^2 w)                                      23.  [(xy)^(-1)/(〖(x〗^(-2) 〖y^(3))〗^3 )]^(-1)

5.  (x^2 y^4 z)^5                                                  24.  (∛(x^2 ))^(1/2)

6.  (a^2/bc)^3                                                      25.  ((8^(5/3)×8^(-1/3))/8^(1/3) )^2

7.  10(x^2 y^3 )^5

8.  ((x^2 y^3 wt^3)/(3pq^2 ))^5

9.  (3x^5 )^2 (2x^3 )^3

10. ((xyz^2)/7a)^2

11. 12x^5÷4x^3

12. 6x^4× 6x^3

13. x^5  × x^0  × y^(-5)

14. x^(-5)/y^(-5 )

15. (15a^(-1) b^2 c^(-3))÷3d

16. (x^(-2) y^(-3))/t^(-2 )

17. (4x^(-3) y^5 )(-2x^2 y^3 )

18. x^(-3)/x^3

19. a(5a^2 b^(-3) )^2