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### Decimal Exponents

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Date: Wed, 02 Nov 1994 14:36:16 -0800
From: Caryl Lynn Segal
Subject: problem

Hope you'll help a graduate student trying to understand an example in an
economics textbook.  This involves food production and the equation is:

Q = 100(K[superscript.8]L[superscript.2])

K involves machines and L involves labor.

The diagram shows Q(output) as 13,800
L as 760
K as  90

I have been unable to figure out how to make the equation reveal those

P.S. I know this is set up for K-12 but am hoping you are not too busy to
help out someone beyond this level
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Date: Wed, 2 Nov 1994 15:40:28 -0500
From: Melissa D. Binde
Subject: Re: problem

kind of economics (micro, macro, etc.) is this?  I'm not an expert in
economics, but perhaps some of the other "doctors" will be able to help you
later this evening.

>P.S. I know this is set up for K-12 but am hoping you are not too busy to
>help out someone beyond this level

We'll do our best!

Melissa
_____________

From: Dr. Ethan
Subject: Re: problem
Date: Wed, 2 Nov 1994 15:50:06 -0500 (EST)

760 to the .8 power times 90 to the .2 power times 100 equals 13789.8,
which rounds to 13800 so I think we are just dealing with multiplication
and exponents.
Ethan Doctor-On Call
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Date: Wed, 02 Nov 1994 17:24:42 -0800
From: Caryl Lynn Segal
Subject: Re: problem

I guess what I'm trying to discover but didn't explain is how you get from
K.8 power to learn it is 90. My problem is being unable to figure out how
to work with .8 power or .2 power etc.
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```
Date: Wed, 2 Nov 1994 18:28:44 -0500
From: Gabe Farmboy Cavallari
Subject: decimal exponents

Here's a possible explanation.  A decimal exponent could also be thought of
as a fractional exponent, so .8 would equal 4/5.  The fractional expression, at
least for me, is easier to use, because raising a number to the 4/5's power
is like raising that number to the 4th power and then taking the 5th root.
For any number raised to the m/nth power, raise the number to the mth (from
the numerator)power and take the nth (from the denominator) root.

As a more familar example, consider the square root.  The exponential
equivalent would be to raise a number to the 1/2 power, or equivalently
the .5 power.

Hope this helps.

Gabe Cavallari, math doc

Date: Wed, 2 Nov 1994 18:34:21 -0500
From: Heather Stickney
Subject: Re: problem

From what you've told us, you can't do the problem.  What I'm thinking is
that this equation is perhaps a standard equation to use to determine
output when you know K and L, or to determine L when you know K and
how much output you want, or vice-versa.  I think the diagram is just showing
an example of how they used the equation.  When K=90 and L=760, there is
an output of 13,800.  You are not expected to use the equation to determine
the values of all the variables at the same time. If this doesn't make
sense, or I've misunderstood what you are asking, write back.>

Date: Wed, 2 Nov 1994 22:46:21 -0500 (EST)
From: "Michael W. S. Morton"

I think there may be another way of looking at this problem (that may be
more in depth than you care to go, or even want to think about.   :)  Having
taken a little econ, I know there are such graphs as isoquants, that is,
graphs where the output of a firm stays constant as you move along the
curve, varying K and L.  These curves show the relationship between
capital (K) and labor (L) at different points, holding the output (Q)
constant.  So, say.....

Q = 13800
The function then looks like:
.8    .2
13800 = 100 * K   * L

.8    .2
138 = K   * L

.8         .2
K  = 138 / L

This function can be graphed (most preferably by a computer).  It
represents the way in which capital and labor can be interchanged at a
constant output of 13800 units.  The slope of this line at any given
point is equal to the Marginal Rate of Technical Substitution (MRTS) at that
point.  More curves can be put on this graph by letting Q = different values.

This may be going further than you wanted to, or maybe it helps.
If you have any questions, just lemme know!

-MORTON, A math Doctor
w/a little econ :)
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Associated Topics:
High School Exponents

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