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
3. Application of Quadratic Equations

LAWS OF INDICES

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

1. aⁿ x a^m = a^n+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 ÷ a^m = a^n-m .

a⁵ ÷ a⁴ = a^5-4 = a¹

Also, a⁵ ÷ a⁴ = = a

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

1. (aⁿ)^m= a^n×m = a^mn

(a³)² = a^3×2 = a⁶

1. a⁰ =1

a⁵ ÷ a⁵ = a^5-5 = 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 = a^5-6 = 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) 32^1/5 (iv) 216 ÷ 3⁴

Solution:

1. (a⁵ x a⁶) /a⁵ = a^{3 +6 -5} = a^9-5 = a⁴
2. (2⁶÷ 2⁷ x 2⁴)^1/3 = (2^6-7+4)^1/3

Re arrange the indices

(2^6+4-7)^1/3 = (2³)^1/3 = 2

iii. 32^1/5 = 32^1/5 but 32 = 2⁵

(2⁵) ^1/5 = 2^{5x 1/5} = 2

iv. 216 ÷ 3⁴

216 = 2³ x 3³

2³ x 3³ ÷ 3⁴ = 2³ x 3^3-4 = 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) 8^2/3 (d) (x³)^-2/3 (e) (4³)⁵

1. Evaluate each of the following

(f) (625)^-1/4 (g) 64^2/3 (h) 9^1/3 x 9^1/6 (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 8^x = 0.25

Solution:

8^x = (2³)^x = 2^3x

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

5^10x = 5⁴

10x = 4

X = 4/10 0r 2/5

CLASS ACTIVITIES:

Solve for x in the following equations

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

SUB TOPIC: APPLICATION OF INDICES LEADING TO QUADRATIC EQUATION

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

1. 5^2x -30 x 5^x + 125 =0

Solution:

Re-write the equation

(5^x)² -30 x 5^x + 125 =0

Let 5^x = 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 =5^x

Therefore 5^x = 5¹, then x = 1

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

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

2^2x + 4(^2x) – 32 = 0

(2^2x)² + 4(^2x) – 32 = 0

Let 2^x = y, then

y² + 4y – 32 = 0

(y + 8)(y – 4) = 0

y = 4 or -8

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

Therefore x = 2

1. Solve for x in the equation

3^2(x-1) – 8(3^(x-2)) = 1

Solution:

Re write the equation

3² x 3^-2 – 8 x 3^x x 3^-2 – 1 = 0

3^2x x 1/3² – 8 x 3^x x 1/3² – 1 = 0

Multiply the equation by 3²

3^2x – 8(3^x) – 3² = 0

Lep p = 3^x

P² – 8p – 9 = 0

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

P = 9 or -1

Recall that p = 3^x

3^x = 3², 3^x = -1 has no solution

x = 2

CLASS ACTIVITIES:

Solve the following equations

1. 2^2x – 5(2^x) + 4 = 0
2. 3^2x+1 + 26(3^x) – 9 = 0
3. 2^2x – 6(2^x) = -8
4. 7^2x – 2 x (7^x) = -1
5. 2^x+3 – 15 = 21^1-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 343^2/3 (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 3^2x – 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 125^x-1 = 25^2x-3
4. If 9^2x+1 = 81^x-2/3^x, what is x?
5. Find the value of x satisfying
6. 3^2x – 30(3^x) + 81 = 0

WEEKEND ASSIGNMENT:

Simplify the following

1. If =
2. Solve
3. If

KEY WORDS:

• INDEX (plural INDICES)
• INDEX FORM
• BASE