Page 19 - Schaum's Outline of Theory and Problems of Applied Physics

4 USEFUL MATH [CHAP. 1

SOLVED PROBLEM 1.8
Solve the equation
16x − 2
= 3x
8
for the value of x.

Cross multiply 16x − 2 = 8(3x) = 24x
Shift −2 and 24x 16x − 24x = 2
Combine like terms −8x = 2
1
Divide both sides by −8 x =− =−0.25
4
SOLVED PROBLEM 1.9
Solve the equation

4x − 35
= 9(1 − x)
3
for the value of x.

Cross multiply 4x − 35 = (3)[9(1 − x)] = 27(1 − x)
Multiply the right-hand side 4x − 35 = 27 − 27x
Shift −35 and −27x 4x + 27x = 27 + 35
Combine like terms 31x = 62
62
Divide both sides by 31 x = = 2
31

EXPONENTS
There is a special shorthand way of expressing a quantity that is to be multiplied by itself one or more times. In
this scheme a superscript number, called an exponent, indicates how many times the self-multiplication is to be
carried out, as follows:
a = a 1 (a)(a) = a 2 (a)(a)(a) = a 3 and so on

2
3
The quantity a is read as “a squared” because it is equal to the area of a square whose sides are a long, and a is
read as “a cubed” because it is equal to the volume of a cube whose edges are a long. For an exponent n greater
5
n
than 3, we read a as “a to the nth power” so that a is “a to the fifth power.” The product of two powers of the
m
m
n
n
same quantity, say a and a , is that quantity raised to the sum of the two exponents: (a )(a ) = a n+m . Thus
2
7
5
(a )(a ) = a .
Reciprocal quantities are expressed according to the above scheme but with negative exponents:
1 −1 1 −2 1 −3
= a = a = a and so on
a a 2 a 3
n
n
0
In general, 1/a = (1/a) = a −n . A quantity raised to the zeroth power, a for instance, is always equal to
1
0
0
−1
1: a = 1. To see why, we note that a/a = 1 can also be written a/a = (a )(a ) = a 1−1 = a .
It is not necessary that an exponent be a whole number. A fractional exponent signifies a root of a quantity.
√
The “square root of a,” customarily written a, is that quantity which, multiplied by itself once, is equal to
√ √ √ 1/2 1/2 1/2
a: ( a)( a) = a. Using exponents, the square root of a is written as a = a , because (a )(a ) =
1
a 1/2+1/2 = a = a. In general, the nth root of any quantity is indicated by the exponent 1/n.