Reviewing Indices
Clement Radcliffe, Contributor

Last week we reviewed Indices and several points were presented for your information. I ask that you look carefully at these and then join me in checking the homework.

Simplify the following:

1. a) 4x³ x 5x²y x 2xy³

b) 9x-4y²÷ 3x³y-5

Solution

a) 4x³ x 5x²y x 2xy³ = 4 x 5 x 2 x x³ x x² x x x y x y³
As multiplication of the above involves adding indices
ie. = 40x6 x4

B) 9x-4 y²÷ 3x³y-5 = (9 ÷ 3) x (x4÷ x³) x (y² ÷ y-5)
As division of the above involves subtracting of indicies:
ie. 3 x x-4-³ y² -5 = 3 x-7 y7

2. Find the values of:

a) 49 to the 1/2 power
b) 27 to the -2/3 power
c) 81 to the 1/4 power

Solve the equation

3. 4³ x-¹= 64 x 4 to the x power

Solution

Expressing all terms as a power of 4:

ie. 4³x-¹ = 4³ x 4x = 4³ - x

Since all terms are expressed in terms of base 4, equating the indices =

3x - 1 = 3 + x ie. 2x = 4 ie. x = 2	Transposing

I expect that you experienced no difficulty so we can now proceed to review Ratio.

Ratio

If two values are in the ratio 1:3, then each represents respectively the fraction 1/4 and 3/4 of the whole.

(1) DECIMAL PLACES

Numbers may be expressed correct to a specified number of decimal places as in the case of the following:

Express 96.31706

(i) Correct to 2 decimal places.
(ii) Correct to 3 decimal places.

Solution

96.32

(Start by looking first at the number which is holding the third place after the decimal point. it is 7. Since 7 is more than 5, 1 is added to the number 1, the number which comes two places after the decimal point.)

N.B. If it were 5 instead of 7, 1 would also be added to the number 1.

(ii) 96.317 {N.B. as 0 holds the fourth place, the 7 remains unchanged.}

(2) SIGNIFICAN FIGURE

The degree of accuracy to which a value is required may be determined by the number of figures in the value. For example, a value expressed correct to two significant figures may be in the form of 24, 1200km or 0.036 litres.

Example

Express 259.163 correct to

(i) 3 Significant figures
(ii) 4 Significant figures

Solution

259 (The number holding the fourth place is 1 so the 8 remains unchanged.)

N/.B. 259.0 is INCORRECT

(ii) 259.2 (N.B. Since 6 > 5, then 1 is added to the 1 similar to the method above.)

Home Work

1. An estate valued at $75,000 is divided among three daughters in the ratio 5:8:2 respectively. Calculate the amount received.

2. Find the following numbers correct to 2 decimal places

a) 5.126

b) 0.085

c) 3.999

3) Divide 41 by 15. Give your answer to 3 decimal places.

4) Express the number 105.8054 corrrect to the number of Significant Figures stated below.

a) 6	b) 4	c) 2
d) 5	e) 3	f) 1

(3) STANDARD FORM

This is a very effective means, especially for very large or very small values. The Standard Form is A x 10n, where A is a number 1 and 10 and n, the power of 10, is an integer (positive or negative whole number, or zero.)

EXAMPLE

Express 3,715,382 in Standard Form.

Since the Standard Form is A x 10n, using the above definition of A and n, then A is 3.72.

It should be noted that CXC accepts values expressed correct to three significant figures. The other figures are simply ignored.

In the number 3,715,382, since the decimal place is after the 2, then it is moved 6 places to the left, to between 3 and 7, consistent with the definition of A, therefore n = 6.

The Standard Form is therefore 3.72 x 106. The answer is (c).

* Clement Radcliffe is principal of Glenmuir High School in Clarendon.