Review of indices, approximation
Clement Radcliffe, Contributor

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

1. Simplify the following

(a) 4x³ X 5x²y X 2xy³

(b) 9x - 4y² / 3x³y - 5

Solution

(a) 5a³b X 4a²b X 7ab³ = 5 x 4 x 7 x a3 x a2 x a x b x b x b3

As multiplication of the above involves adding indices:

5a³b X 4a²b X 7ab³ = 140a6b5

(b) 9x - 4y² / 3x³y - 5 = (9 / 3) x (x-4 / x3) x (y2 / y-5)

As division of the above involves subtracting of indices:

= 3 x x-4 -3 x y2- -5 = 3x -7y7

2. Find the values of

(a) 64-5/6

(b) 27-2/3

(c) 813/4

Solution

(a) Given 64-5/6, we express 64 in terms of base 5. 125-2/3 = (53) -2/3. In this case, you multiply indices.

= 5-2 =1/25

(b) 27-2/3 = (25) -3/5 = 2 -3

= 1/8

(c) 813/4 = (34) 1/4 = 31 = 3

3. Solve the following equation for x

9²x = 1/27

Solution

(3²)²x = 27^-1 = (3³)^-1

34x = 3^-3

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

4x = -3

x = -3/4

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

Ratio

If two values are in the ratio 3:5, then each represents, respectively, the fraction of 3/8 and 5/8 of the whole.

Proof

In this case, the whole is taken as 3 + 5 = 8.

The fractions are 3/8 and 5/8.

It is vital for you to be able to convert ratios to fractions in all cases.

Example

A number is divided in the ratio 2:3. What fraction does the smaller ratio represent?

As the number is divided into the ratio 2:3, then 2 + 3 = 5

The fractions are 2/5 and 3/5

The answer is 2/5

Example

A sum of money is to be divided among A, B and C in the ratio 3:4:5. The largest portion amounts to $1,800.

Calculate:

(a) The total sum of money to be shared

(b) A's share

Since the money is shared in the ratio 3:4:5 and the whole is represented by 3 + 4 + 5 = 12, the respective portions are:

A = 3/12 or 1/4

B = 4/12 or 1/4

C = 5/12

If the largest share = $1,800, then this represents C's share, the total sum is 1,800 x 12/12.

The total sum is $4,320.

A's share represents 1/4 of the total. This is equal to 1/4 x $4,320 = $1,080

A's share is $1,080

Finally, we will now briefly review aspects of approximation.

Approximation

This topic highlights the various degrees of accuracy to which a value may be expressed. While counting always gives an accurate assessment, it is measurement which lends itself to approximation, depending on the nature of the instrument used. For example, an electronic balance can measure the weight of a sample to three or more decimal places while this degree of accuracy is not always required. You, therefore, have the option of giving a value to the degree of accuracy you require.

The three methods which are usually used at this level are:

• Decimal places
• Significant figures
• Standard form

(1) Decimal places

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

Express 46.42806

(i) Correct to 2 decimal places.

(ii) Correct to 3 decimal places.

Solution

(i) 46.43 (Start by looking first at the number which is holding the third place after the decimal point. It is 8. Since 8 is more then 5, 1 is added to the number 2, the number which comes two places after the decimal point.)

NB If it were 5, instead of 8, holding the third place after the decimal, 1 would also be added to the number 2.

(ii) 46.428

(2) Significant 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 24,1200 km or 0.036 litres.

Example

Express 259.163 correct to:

(i) 3 significant figures

(ii) 4 significant figures

Solution

(i) 259 (The number holding the fourth place is 1 so the 9 remains unchanged.)

NB 259.0 is incorrect

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

(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.

a) 372 x 10⁶

b) 3.72 x 10^-6

c) 3.72 x 10⁶

d) 3.72 x 10⁷

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 the Caribbean Examination Council 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).

Homework

1. An estate valued at $90,000 is divided among three daughters in the ratio 5 : 8 : 2, respectively. Calculate the amount each received.

2. Find the following numbers correct to 2 decimal places.

a) 4.028

b) 0.055

c) 6.999

3. Divide 56 by 13. Give your answer to 3 decimal places.

4. Express the number 105.7064 correct to the number of significant figures stated below.

a) 6

b) 4

c) 2

d) 5

e) 3

f) 1

Clement Radcliffe is the principal of Glenmuir High School in May Pen.