Lesson Notes By Weeks and Term - Senior Secondary School 1

SUBJECT: MATHEMATICS

CLASS:  SS 1

DATE:

TERM: 1st TERM

REFERENCE BOOKS

• New General Mathematics SSS 1     M.F. Macrae et al 

 
WEEK SIX

TOPIC: INDICES

CONTENT:

• Laws of Indices.

• Negative and Fractional Indices.

• Solving Equation Under Indices.

LAWS OF INDICES

 1st to 4th laws for all values of a , b and x≠ 0

1. Xa x Xb = Xa+b
2. Xa ÷ Xb = Xa-b
3. X0 = 1
4. X-a=  1

Xa

Examples:

Simplify

1. 105 X 104        2. a3 X a4         3. m8 ÷ m5         4.  24x6 ÷ 8x4      5.   198 ÷ 198

Solutions

1. 105 X 104 = 105+4 =109
2. a3 X a4 = a3+4 =a7
3. m8 ÷m5 = m8-5 = m3
4. 24x6 ÷ 8x4 =      24x6 =    3x6-4   =3x2

                           8x4

1. 198 ÷ 198 = 198-8 = 190 =1

Evaluation

Simplify

1. 6 x Z0     (b) 4-3    (c) Z3 x (⅙)1    (d) r x r x r x r-5

PRODUCT OF INDICES

(Xa)b = Xaxb = Xab

Examples

Simplify

1. (X2)3             2. (Y4)2            3.   (3-2)-3      4. (-3d3)2      5. a6(-a)-4

Solutions

1. (X2)3 = X2X3 = X6
2. (Y4)2 = Y4X2 = Y8
3. (3-2)-3 = 3-2 X -3=3+6

             =36 =3 X 3 X 3 X 3 X 3 X3

        =27 X 27

          = 729

1. (-3d3)2 = (-3)2 X (d3)2

       = -3 X -3d6 = 9d6

1. a6(-a)-4 =  a6 X   1

  (-a)4

               =             a6

              (-a) X(-a) X (-a) X(-a)

    =  a6

              a4

              = a6 - 4

              = a2

EVALUATION

Simplify

1. (h4)-5        2.  (-4u2v)3     3.  (-x3)2÷ x4     4.  – (d2) ÷ d4 x –d     5.  (-c)2 X (c)4 ÷ (-c3)

FRACTIONAL INDICES

X1/a and X a/b

    X    is short for the square root of x

√X  X  √ X = X

Let √x = xp

Then 

Xp X xp =√ x X √ x= x1

By equating the indices

2p = 1   ,     P =½

Thus √x – x1/2  =  3 x

Similarly, 3  x is the short for the cube root of x e.g3 8= 2. Since 2 X 2 X 2 =8

And 3√-27 = -3

 Since (-3 ) X (-3)  X (-3 )  = -27

3√x X 3√x X 3√x = x

i.exq X xq X xq = x1

x3q = 1

Equating the power 

3q=1

q= ⅓

thus3√x = x⅓

In general x1/a =a√x

Also  x2/3 = x2 X 1/3= (x2)1/3

=3√x2

OR

X2/3 = (x2 x 1/3)= (x1/3)2

= (3√x)2

In general 

Xa/b = b√xaor (b√x)a

Examples

Simplify

1. 8-2/3    2.  4 1/6 X 4 1/3     3. (16/81)-3/4

4.√72a3b-2/2b5b-6

Solution

1. 8-2/3 =    1

             82/3

 =          1

            (3√8)2

    =  1      =    1

            (2)2          4

1. 41/6 X 41/3 = 41/6÷ 1/3

=  43/6 = 41/2

=√4 = 2

1. (16/81)-3/4 =         1

                        (16/81)3/4

=   1

( 4√16/81)3

=    1

   (2/3)3

= 1 ÷ (2/3)3

=1 ÷8/27 = 1 X 27/8 = 27/8

1.  72a2b-2    =     (72a3b-2)1/2

2b5b-6                2a5b-6

=  72 X a3 Xa-5 X b2 X b-6

     2

= √36a3-5 X b-2-(-6)

= √36a-2 X b-2+6

= √36a-2 X b4

= √36 X (a-2 X b4)1/2

=6 X a-2 X1/2 X b4x1/2

=6a-1 X b2 =6 X 1X b2

a

=6b2

a

EVALUATION

Simplify 1. (125)-1/3    2.   (18/32)-3/2      3.(3√4)1.5      4.64-5/6      5.   √1  9/16

SOLVING EQUATION WITH INDICES

Solve the following equations:

1. 2r-3 = -16
2. 5x = 40  x-1/2

5      5

1. 4c-1 =64

Solutions

1. 2r-3 = -16

Divide both sides by 2

2r-3 = -16

2          2

r-3 = -8

1   = -8

r3       1

-8r3 = 1 X 1

r3 = - 1

         8

Take cube root of both sides

3√r3 = 3 - 1

             -8

r = -1

       2

1. 5x = 40x-1/2

5        5

x= 8x-1/2

x=  8 x  1

x1/2

Cross multiply 

xX x1/2 =8

x1 X x1/2 =8

x1+1/2 =8

x3/2 =8

i.e (√x)3= 8

raise both sides by power  2/3

(x3/2) X 2/3 = (8)2/3

X1= (3√8)2

X= (2)2

X= 4

1. 4c-1 =64

Change both sides to the same base

4c-1 = 43

Equate the powers

c-1 = 3

c = 3 + 1

c = 4

EVALUATION

Solve the following equation

1. a2/3= 9      2.  2x3 = 54

GENERAL EVALUATION

1. If 92x +1 = 81x-2 find x

3x

1. Solve 9x-1 = 27x+1
2. Simplify 3 72p-3q-7

                    9p9q5

READING ASSIGNMENT

NGM SSS page 18, exercise 1d numbers 21-50.

WEEKEND ASSIGNMENT 

1. Simplify 33 X 6-3 X 25                                                   (a) 0             (b) 1        (c) 2       (d)4           (e) 12
2. Calculate the value of (27/125)1/3 X (4/9)1/2 (a)12/25      (b) 2/5    (c) 3/5   (d) 9/10    (e) 10/9
3. If 5p-3 = 8 X 5-2, find the value of p                   (a) 8/125    (b) 2/5   (c) 4/5   (d) 8/5       (e) 5/2
4. If x2 = 811½, x = ----------                                       (a) 3             (b) 9       (c)18     (d)27          (e) 54
5. Simplify (x1⅓)3 x 1 (a)1              (b) x1        (c) x3       (d) x0           (e)2

x4

THEORY

1. Evaluate 9½ X 27 2/3

                   641/3

1. Solve the following equations

1. Y-2 = 9       (b) (2s)1/2 = 9    (c) 2n-1 = 16