|
Calculating
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 again and then join me in checking the
homework. Simplify
the following: 1.
(a) 2x³
x 3x²y
x 5xy³
(b) 18x-4y-4
÷ 3x³y-5 Solution
(a)
2x³ x
3x²y x
5xy³ =
2 x 3 x 5 x x³
x x² x
x x y x y³ As
multiplication of the above involves adding indices: ie.
= 30x6 y4 (b)
18x-4y²÷
3x?y-5 = (18
÷ 3) x (x-4÷
x³) x
(y²÷
y-5) As
division of values with the same base involves subtracting the indices: ie.
= 6 x x-4-3
x y²--5
= 6 x x-7 y7
= 6x-7y7 2.
Find the values of: (a)
125-2/3 (b)
32-3/5 (c)
811/4 Solution
(a)
Given 125-2/3
we express 125 in terms of base 5. ie.
125-2/3 = (5³)-2/3.
In this case, you multiply indices. =
5-2 = 1/25
(b)
32-3/5 = (25)-3/5
= 2-3 =
1/8 (c)
811/4 = (34)1/4
= 3¹ =
3 3.
Solve the equation 24x
= 64 Solution
Since
24x = 64 Then
express 64 in terms of base 2. ie.
24x = 26
Equating indices as both are with respect to base 2.
ie. 4x = 6
ie. x =
6/4 = 3/2
ie. x = 3/2 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
of 1/4 and 3/4 of the whole. Proof
In
this case, the whole is taken as 1 + 3 = 4. ie.
The fractions are 1/4 and 3/4. It
is vital for students to be able to convert ratios to fractions in all cases.
Example
A
number is divided in the ratio 3:5. What fraction does the smaller ratio represents?
As the
number is divided into the ratio 3:5, then 3 + 5 = 8 ie.
The fractions are 3/8 and 5/8 ie.
The answer is 3/8 Example
A
sum of money is to be divided among A, B and C in the ratio 2:3:5. The largest
portion amounts to $2,400.
Calculate:
(a)
The total sum of money to be shared (b)
A's share Since
the money is shared in the ratio 2:3:5 and the whole is represented by 2 + 3 +
5 = 10, the respective portions are as follows:
A
= 2/10 or 1/5
B = 3/10
C = 5/10 or 1/2
If
the largest share = $2,400, then this represents C's share which is half of the
total sum. ie.
The total sum is $4,800. A's
share represents 1 of the total. This is equal to 1 x $4,800 = $960 5
5
ie. A's share is $960 Finally
we will 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 of 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: 1.
Decimal Places
2.
Significant Figures
3.
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
54.31706 (i)
Correct to 2 decimal places.
(ii)
Correct to 3 decimal places. Solution
(i)
54.32(Start by looking first at the number which is holding the third place after
the decimal point. It is 7. Since 7 is more then 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, you would also add 1 to the number 1. (ii)
54.317 (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, 1200km 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.) 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.)
Common
Error Express
37 564 correct to three significant figures. There
are some students who may consider the answer to be 376 instead of 37 600. Please
be reminded that both values, the number given and its approximation, MUST be
APPROXIMATELY the same. Home
Work 1.
An estate valued at $75,000 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)
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 correct to the number of significant figures stated
below. a)
5
b) 3
c) 1 Clement
Radcliffe teaches at Glenmuir High School. | 
