Rules of Exponent ManipulationDate: 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/ |
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