Page 20 - Schaum's Outline of Theory and Problems of Applied Physics

CHAP. 1] USEFUL MATH 5

Some rules concerning exponents and roots are the following:
a a
n n
n
n n
nm
n m
(ab) = a b (a ) = a =
b b n
a a n 1/m n/m
1/m 1/m
1/m 1/m
1/m
(ab) = a b = (a ) = a
b b 1/m
SOLVED PROBLEM 1.10
Examples of exponents:
2 −3
2 −4
3 −1/3
a a = a 2−3 = a −1 (a ) = a (2)(−4) = a −8 (a ) = a (−1/3)(3) = a −1
6 1/2
−1 −4
−3 −2
a a = a −1−4 = a −5 (a ) = a (−3)(−2) = a 6 a a = a 6+1/2 = a 13/2
−2 4
6 1/2
(a ) = a (−2)(4) = a −8 (a 1/2 6 (6)(1/2) = a 3 a(a ) = a (1/2)(6) = a 3
) = a
SOLVED PROBLEM 1.11
2
The area A of a circle whose radius is r is given by A = πr . Find the area of a circle whose radius is
4.00 meters (m).
The value of π is 3.14159 .... Many calculators have a key for this quantity. The result is
2
2
2
A = πr = π(4.00 m) = π(4.00 m)(4.00 m) = 50.3m .
2
The symbol “m ” stands for “square meters.”
SOLVED PROBLEM 1.12
2
Find the radius of a cylindrical tank whose cross-sectional area is 5.00 m .
2
2
We begin by solving A = πr for r and then substitute A = 5.00 m to find the value of r:
A = πr 2
A
2
r =
π

A 5.00 m 2 √
2
r = = = 1.59 m = 1.26 m
π π
√
2
Note that m = m.
POWERS OF 10
Very small and very large numbers are common in science and engineering and are best expressed with the help of
powers of 10. A number in decimal form can be written as a number between 1 and 10 multiplied by a power of 10:
834 = 8.34 × 10 2 0.00072 = 7.2 × 10 −4

6
The powers of 10 from 10 −6 to 10 are as follows:
0
10 = 1 = 1 with decimal point moved 0 places
10 −1 = 0.1 = 1 with decimal point moved 1 place to the left

10 −2 = 0.01 = 1 with decimal point moved 2 places to the left
10 −3 = 0.001 = 1 with decimal point moved 3 places to the left