# Laws of Indices

## Further Mathematics SS 1 FIRST TERM

WEEK ONE

LAWS OF INDICES:

CONTENT:

1. Laws of Indices
2. Application of Indices Linear Equations

LAWS OF INDICES

There are laws governing the use of indices. These are useful in other subjects. They are;

1. an x am = an+m

Therefore

a3 x a3 = a x a x a x a x a x a = a6 i.e a3+3 = a6

In general, when multiplying indices with same base, you add the power

1. an ÷ am = an-m .

a5 ÷ a4 = a5-4 = a1

Also, a5 ÷ a4 = = a

In general, when dividing indices with the same base, you subtract the power

1. (an)m= an×m = amn

(a3)2 = a3×2 = a6

1. a0 =1

a5 ÷ a5 = a5-5 = a0

Also, a5 ÷ a5 = = 1

This implies that anything to power zero is equal to 1, i. e 50 = 1, 20 = 1

1. a-n= 1/an

Consider a5 ÷ a6 = a5-6 = a-1

But a5÷ a6 = =

a-1 =

In the same way a-2=,

1. a n/m =n = )

Consider (8) 1/3 = (23) 1/3 = (23) 1/3 = 2

= 2,

Example1:

(i) (a5 x a6) /a5 (ii) (26 ÷27 x 24)1/3 (iii) 321/5 (iv) 216 ÷ 34

Solution:

1. (a5 x a6) /a5 = a3 +6 -5 = a9-5 = a4
2. (26÷ 27 x 24)1/3 = (26-7+4)1/3

Re arrange the indices

(26+4-7)1/3 = (23)1/3 = 2

iii. 321/5 = 321/5 but 32 = 25

(25) 1/5 = 25x 1/5 = 2

iv. 216 ÷ 34

216 = 23 x 33

23 x 33 ÷ 34 = 23 x 33-4 = 23 x 3-1

23/3 0r 8/3

Example2:

Simplify

Solution

= = =

CLASS ACTIVITIES:

1. Evaluate each of the following

(a) 80 (b) 5-1 (c) 82/3 (d) (x3)-2/3 (e) (43)5

1. Evaluate each of the following

(f) (625)-1/4 (g) 642/3 (h) 91/3 x 91/6 (i) 36 ÷ 37 x 22 (j) (1000)-5/3

SUB TOPIC: APPLICATION OF THE LAWS OF INDICES.

Examples:

1. Solve (1/2)x = 8

Solution:

(1/2)x = (2-1)x = 2-x

8 = 23 since we have same base, then –x = 3.

Multiply the equation by (-1)

X = -3

1. Solve the equation 8x = 0.25

Solution:

8x = (23)x = 23x

0.25 = 25/100 = ¼ = (1/2)2 = (2-1)2 = 2-2

23x = 2-2

3x = -2

X = -2/3

1. Solve for x in the equation: (0.25)x+1 = 16

(1/4)x+1 24

(2-2)x+1 = 24

2-2x-2 = 24

Equate the power

-2x – 2 = 4

-2x = 4 + 2 = 6

X = -6/2 = -3

X = -3

If 10-x = 0.001. what is the value of x?

0.001 = 10-3

–x = -3

x = 3

1. If 25(5x) = 625, what is x.?

(52)5x = 54

510x = 54

10x = 4

X = 4/10 0r 2/5

CLASS ACTIVITIES:

Solve for x in the following equations

1. 3x = 81
2. 2x = 32
3. 9x = 1/729
4. 25(5x) = 625
5. 2x x 4-x = 2

Some exponential equations will lead to quadratic equations as you will see in the following examples.

1. 52x -30 x 5x + 125 =0

Solution:

Re-write the equation

(5x)2 -30 x 5x + 125 =0

Let 5x = p, then

P2 – 30p + 125 = 0

Solve for p by factorization

(p-5)(p-25) = 0

P – 5 = 0 or p – 25 = 0

Then p = 5 or 25,

Recall that p =5x

Therefore 5x = 51, then x = 1

or 5x = 25 this means that 5x = 52, x = 2

Solve the equation 22x + 4(2x) – 32 = 0

22x + 4(2x) – 32 = 0

(22x)2 + 4(2x) – 32 = 0

Let 2x = y, then

y2 + 4y – 32 = 0

(y + 8)(y – 4) = 0

y = 4 or -8

Then 2x = 22 or 2x = -8. But this (2x = -8) has no solution

Therefore x = 2

1. Solve for x in the equation

32(x-1) – 8(3(x-2)) = 1

Solution:

Re write the equation

32 x 3-2 – 8 x 3x x 3-2 – 1 = 0

32x x 1/32 – 8 x 3x x 1/32 – 1 = 0

Multiply the equation by 32

32x – 8(3x) – 32 = 0

Lep p = 3x

P2 – 8p – 9 = 0

(p-9)(p+1) = 0

P = 9 or -1

Recall that p = 3x

3x = 32, 3x = -1 has no solution

x = 2

CLASS ACTIVITIES:

Solve the following equations

1. 22x – 5(2x) + 4 = 0
2. 32x+1 + 26(3x) – 9 = 0
3. 22x – 6(2x) = -8
4. 72x – 2 x (7x) = -1
5. 2x+3 – 15 = 211-x

PRACTICE EXERCISE:

Objective Test:

Choose the correct answer from the options

1. Simplify (105)0 (a) 0 (b) 1 (c) 5 (d) 3 (e)-1
2. Evaluate 3432/3 (a) 7 (b) 49 (c) 343 (d) 3 (e) 9
3. Simplify (28 x 4-3) / 26 (a) 1/16 (b) 16 (c) 220 (d) ¼ (e) 2/4
4. Solve the equation 3-x = 243, x = ? (a) 5 (b) 3 (c) 4 (d) -5 (e) -3
5. Solve the equation 32x – 9 = 0, x = ? (a) ±3 (b) 2 (c) -2 (d) 1 (e) 0

Essay questions:

1. Simplify (a) 4
2. (1/3)4 x 36 ÷ (2/3)2
3. Solve for x, if 125x-1 = 252x-3
4. If 92x+1 = 81x-2/3x, what is x?
5. Find the value of x satisfying
6. 32x – 30(3x) + 81 = 0

WEEKEND ASSIGNMENT:

Simplify the following

1. If =
2. Solve
3. If

KEY WORDS:

• INDEX (plural INDICES)
• INDEX FORM
• BASE
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