Indices
Clement Radcliffe, Contributor

Students of Vauxhall Comprehensive High School perform a skit during World Wetlands Day 2006 celebrations at the UWI marine lab at Port Royal on Thursday, February 2. - Andrew Smith/Photography Editor

We will begin this week's lesson with a review of the homework given last week.

1. Mr. Mitchell deposited $40,000 in a bank and earned simple interest at 7% per annum for two years. Calculate the amount he will receive at the end of the two-year period.

SOLUTION

Since Mr. Mitchell deposited $40,000 at 7% per annum for two years:

The amount received after two years is Principal + Simple Interest

= $40,000 + $5,600 = $45,600

2. Mr. Williams bought a plot of land for $40,000. The value of the land appreciated by 7% per each year.

Calculate the value of the land after a period of two years.

SOLUTION

Since Mr. Williams purchased the land for $40,000 and it appreciates by 7% annually: Value after the first year is:

The value after the second year is:

I do hope that you realise the difference between both exercises. In the second case, the increase is compounded. Please note this well.

The lesson will continue with a review of Indices, an aspect of Computation.

INDICES

This is the power of a number. For example, 16 may be expressed in the form 2 to the power of 4, that is 24. In this case, 4 is the index or power of 2.

Example: Express 128 as a power of 2.
As 128 = 2 x 2 x 2 x 2 x 2 x 2 x 2
128 = 27

Expressing numbers in index form is fundamental to solving certain problems.

Points to note:

(a) It is to your benefit to know the value of some whole numbers raised to powers, for example:

Powers of 2 up to 27, for example, 24 = 2, 2³ = 8, 24 = 16, etc.

Powers of 3 up to 5³, for example, 3² = 9, 34 = 81, etc.

(b) Denominator of a fractional power represents root.

For example, 81/3 is another way of writing the cube root of 8, therefore 8³ = 2.

N.B. 82/3 is the square of the cube root of 8. It can be written also in the form
(81/3)²

(c) Any number to the power zero is equal to 1, for example, 4º = Xº = 1.

(d) Negative power represents the reciprocal, for example,

3-² is commonly misinterpreted as -3² = -9. Avoid making this error. Repeating 2-³ = (2-¹)³ =(1/2)³ = 1/8

(e) When we multiply numbers with the same base, we add the indices.

e.g., 2³ x 24 = 2³ + 24 = 27

Can you say why this is so?

It can be shown to be true as follows.

2³ x 24 = 2 x 2 x 2 x 2 x 2 x 2 x 2 = 27

(f)To divide numbers with the same base, we subtract the indices.

e.g., 56÷54 = 56-4 = 5²

NOTE: This may be justified by expanding and dividing.

Similarly X²÷ X5 = X2-5 = X-³

I am sure that you have noted the importance of directed numbers.

(g) In attempting to simplify an expression, it is always necessary to express each term in the form of its smallest factor, for example:

Evaluate: 8² x 45

Given that 8 = 2³ and 4 = 2²

8² x 45 = (2³)² x (2²)³ = 26 x 210 = 216.

Let us apply these to the following examples:

1. 2a²b x -4ab³

(A) -2ab² (B) 6a³b4 (C) -8a³b4 (D) -a-3b4

SOLUTION

2 x -4 = -8, a² x a = a³, b x b³ = b4

The product is -8 x a3 x b4 = -8a3b4

Answer is (C)

2. Simplify: 811/2 x 27-1/2

As 81 = 34 and 27 = 3³, then

811/2 x 27-1/3 = (34)1/2 x (3³)-1/3

As 4 x 1/2 = 2 and 3 x -1/3 = -1

(34)1/2 x (3³)-1/3 = 3² x 3-1 = 3

3. Solve the following equation for x.

92x = 1/27

(3²)2x = 27-1 = (3³)-1

34x = 3-3

Since 4x and -3 are both powers of 3 then

4x = -3
x = -3/4

This topic, as I said before, is a crucial one and I want you to absorb the information given in this lesson. Reinforce the concepts by doing the following for homework.

Simplify the following:

1. (a) 2x³ x 3x²y x 5xy³ (b) 18x-4y²÷ 3x³y-5

2. Find the values of:

(a) 125-2/3 (b) 32-3/5 (c) 811/4

3. Solve the equation 24x = 64

Clement Radcliffe teaches at Glenmuir High School.