Common Mathematical Operations In Chemistry Worksheets With Answers Page 3
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14Appendix A
•
Common Mathematical Operations in Chemistry
A-3
3. An exponential number with a coefficient greater than 10 or less than 1 can be
changed to the standard exponential form by converting the coefficient to the stan-
dard form and adding the exponents:
6
2
6
(26)
8
582.310
changes to 5.823 10
10
5 5.82310
5 5.82310
0.004310
changes to 4.3 10
10
5 4.310
[(3)(4)]
5 4.310
4
3
4
7
Using Exponential Notation in Calculations
In calculations, you can treat the coefficient and exponents separately and apply the
properties of exponents (see earlier section on logarithms).
1. To multiply exponential numbers, multiply the coefficients, add the exponents, and
reconstruct the number in standard exponential notation:
3
5
(35)
8
9
(5.510
)(3.110
) 5 (5.5 3.1)10
5 1710
5 1.710
(9.710
14
)(4.310
) 5 (9.7 4.3)10
[14(20)]
5 4210
5 4.210
20
6
5
2. To divide exponential numbers, divide the coefficients, subtract the exponents, and
reconstruct the number in standard exponential notation:
6
2.6310
2.6
10
(62)
5 0.4510
4
5 4.510
3
5
2
5.8310
5.8
1.7310
25
1.7
10
[(5)(8)]
5 0.2110
3
5 2.110
2
5
8.2310
28
8.2
3. To add or subtract exponential numbers, change all numbers so that they have the
same exponent, then add or subtract the coefficients:
(1.4510
4
) (3.210
3
) 5 (1.4510
4
) (0.3210
4
) 5 1.7710
4
5
4
5
5
5
(3.2210
) (9.0210
) 5 (3.2210
) (0.90210
) 5 2.3210
solving QUAdrAtic EQUAtions
A quadratic equation is one in which the highest power of x is 2. The general form
of a quadratic equation is
2
ax
bx c 5 0
2b 6 "b
where a, b, and c are numbers. For given values of a, b, and c, the values of x that
satisfy the equation are called solutions of the equation. We calculate x with the qua-
dratic formula:
2
2 4ac
x 5
2a
We commonly require the quadratic formula when solving for some concentration in
an equilibrium problem. For example, we might have an expression that is rearranged
into the quadratic equation
2 0.65 6 " 1 0.65 2
2 4 1 4.3 2 1 28.7 2
4.3x
2
0.65x 8.7 5 0
2 1 4.3 2
a
b
c
Applying the quadratic formula, with a 5 4.3, b 5 0.65, and c 5 8.7, gives
2
x 5
The “plus or minus” sign () indicates that there are always two possible values for
x. In this case, they are
x
1.3
and
x
5
5
1.5
In any real physical system, however, only one of the values will have any meaning.
For example, if x were [H
O
], the negative value would give a negative concentra-
3
tion, which has no physical meaning.
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