Topic 1: Equations Involving Exponents

Competencies

·         Solve equations involving exponents

·         Use exponent rules

Suggested ways of teaching this topic: Teacher guided group discussion

Starter Activities

The teacher may start raising brainstorming questions of the following type:

·         “For what value(s) of x will 2^x = 8?”

·         “When do we say 2^x =2^y?

And, let the students come up with answers discussing in pairs or in small groups.

Expected Answers:

·         x= 3 for the first and x = y  for the second

The teacher should summarize students discussion raising the rules for exponents andthe fact that for a > 0, a^x= a^y if and only if x = y.

Lesson Notes

Exponent Rules:

For a, b >0 and x and y are real numbers;

1.      a^x a^y= a^x+y

2.      a^x a^y = a^x-y

3.      (a^x)^y= (a^y) ^x= (a)^xy

4.      a^xx =(ab)^x

^5.        a^x b^x= (ab)^x

Example1: Find the value of x

a)   4^2x = 32

b)   3^x+5  = 27

Solution:

a)      42x = 32   (22)2x = 25 24x = 25 4x = 5 x = 5/4

b)3x+5  = 27 3x+5= 33 x + 5 = 3 x = 3 – 5  x = -2

Example 2: Find the solution to the following equations

a)    (27)(9^{x – 1})  = 81^xb)   2^{5 – x} = (^{x – 3}

Solution:

a)    (27)(9x – 1)  = 81x (33)(32(x-1)) = 34x 33+2x-2=34x 32x+1=34x 2x+1 = 4x -2x = -1 x =1/2

b)25 – x = (x – 3 25-x= 2-2(x-3) 25-x = 2-2x+6 5 – x = -2x +6  x = 1

Example 3: Solve (2t - 3)^-2/3 = -1.

Solution:

Raise each side to the power -3 to eliminate the root and the negative sign in the exponent:

(2t - 3)-2/3	= -1
[(2t - 3)-2/3]-3 = (-1)-3		Raise each side to the -3 power.
(2x - 3)2	= -1	Multiply the exponents:

There is no real number which when squared can give – 1. Thus, the equation has no solution.

Concluding Activities

Make sure that students have got the idea, “To solve equations involving exponents:

·         Use the exponent rules and

·         The fact that a^x =a^y if and only if x = y.”

Practice Exercise

Solve the following equations

a)

b) = 2-x

c)  =  32x