Rules of Exponent Manipulation

Date: Mon, 22 Jan 1996 10:22:40 -0500
From: Anonymous
Subject: Algebra

I have pushed the "I believe button" on this one but want to know the 
reason behind the solution.

The problem is:

f sub r equals the reciprocal of 2 times 3.14 times square root of 
0.02 times 0.18 times 10e-6

or

f sub r equals the reciprocal of 6.28 times square root of 
0.0036 times 10e-6

or

f sub r equals the reciprocal of 6.28 times square root of 
36 times 10e-10

or

f sub r equals the reciprocal of 6.28 times 10e-5 times square root 
of 36

or

f sub r equals 10e5 divided by 37.7

What rule is it to halve the square root of 36 times 10e-10 and 
move it to the left thus becoming 6.28 times 10e-5 times square root 
of 36?

What rule is it to move the 10e-5 from the divisor to the dividend 
making it positive? 

Date: 9/1/96
From: Doctor James
Subject: Re: Algebra

This problem is easier to see when it is translated into some 
internet-math shorthand. In this: 

5 squared, or 5 to the power of 2 = 5^2, and 3 times 4 = 3*4.

Now, the rules of exponents say that:

               (a^b)*(a^c) = a^(b+c),
  [ a to the power of b times a to the power of c equals a to the 
power of b plus c. See how much shorter it is without words! ]

and                (a^b)^c = a^(b*c).

Also they say that

               (a^c)*(b^c) = (a*b)^c.

An example of the last would be 3 to the power of 2 times 4 to the 
power of 2 equals three times four to the power of 2, or

               (3^2)*(4^2) = (3*4)^2 = 12^2

It is important to remember that the square root of 36 = 36^(1/2), and
the reciprocal of 10 = 1/10 = 10^(-1).

So let's write your problem in this way.

f_r = [ 2 * 3.14 * [0.02 * 0.18 * 10^(-6)]^(1/2) ]^(-1)   
   [what a mess!]
f_r = [ 6.28 * [ 0.0036 * 10^(-6) ]^(1/2) ]^(-1)
f_r = [ 6.28 * [ 36 * 10^(-10) ]^(1/2) ]^(-1)             
   [getting better!]

Now this is where we must call upon the rules of exponents. Let's 
just look at the inner square brackets

               [ 36 * 10^(-10) ]^(1/2)

but by the third rule I said above, this is

               (36)^(1/2) * ( 10^(-10) )^(1/2)

by the second rule, this becomes
 
              (36)^(1/2) * ( 10^[-10 * (1/2)] )
or            (36)^(1/2) * ( 10^(-5) )
as -10 * (1/2) = -5.  Then

                 f_r  = [ 6.28 * 36^(1/2) * 10^(-5) ]^(-1).

The answer to your second question is similar.

                 f_r  = [ 37.7 * 10^(-5) ]^(-1)
                      = ( 10^(-5) )^(-1) * 37.7^(-1)
                      = 10^(-5 * -1) * 37.7^(-1)     
[from the second rule]
                      = (10^5) * 37.7^(-1)
                      = (10^5) / 37.7,

which is the answer you believed!

Hope this helped; if not, please email me again.
  
-Dr. James,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/