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August 6th, 2013, 09:00 PM   #1
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Combining equations; exponents and fractions

I am a geologist by training, but either 1.) I seem to have forgotten some of the basics of algebra, or 2.) the publication from which I obtained this equation contains an error (I'm going to go ahead and assume the former).

Please see the attached Word document as a reference for my question.

The publication explains that equations (1) and (2) are combined to obtain equation (3), but equation (4) is the only answer I have been able to come up with.

Am I missing some rule/step here?

Thanks for the help!
Attached Files
 CrushedCellEquations.docx (14.6 KB, 23 views)

 August 6th, 2013, 10:04 PM #2 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,780 Thanks: 970 Re: Combining equations; exponents and fractions Can't reach that link.
August 7th, 2013, 11:09 AM   #3
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Re: Combining equations; exponents and fractions

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 August 7th, 2013, 11:25 AM #4 Senior Member   Joined: Jul 2013 From: United Kingdom Posts: 470 Thanks: 40 Re: Combining equations; exponents and fractions I think it will have to be broken down to get a clean final answer. give me a few minutes. your answer contains too much information.
 August 7th, 2013, 11:37 AM #5 Senior Member   Joined: Jul 2013 From: United Kingdom Posts: 470 Thanks: 40 Re: Combining equations; exponents and fractions a can be changed into: $a=c{ \left( \frac { { D }^{ 2 } }{ { d }^{ 2 } } \right) }-p\left( \frac { { D }^{ 2 }-{ d }^{ 2 } }{ { d }^{ 2 } } \right) \\ =c{ \left( \frac { D }{ d } \right) }^{ 2 }-p\left( \frac { { D }^{ 2 } }{ { d }^{ 2 } } -\frac { { d }^{ 2 } }{ { d }^{ 2 } } \right) \\ =c{ \left( \frac { D }{ d } \right) }^{ 2 }-p\left[ { \left( \frac { D }{ d } \right) }^{ 2 }-1 \right] \\ =c{ \left( \frac { D }{ d } \right) }^{ 2 }-p{ \left( \frac { D }{ d } \right) }^{ 2 }+p\\ ={ \left( \frac { D }{ d } \right) }^{ 2 }(c-p)+p$
 August 7th, 2013, 11:54 AM #6 Senior Member   Joined: Jul 2013 From: United Kingdom Posts: 470 Thanks: 40 Re: Combining equations; exponents and fractions step 3 is unnecessary. could lead you into a world of confusion. this final answer seems ok to me. please see if it works before using it: $r={ \left( \frac { { \left( \frac { D }{ d } \right) }^{ 2 }(c-p)+p }{ { w }^{ q } } \right) }^{ \frac { 1 }{ 1-q } }$
 August 7th, 2013, 01:07 PM #7 Newbie   Joined: Aug 2013 Posts: 7 Thanks: 0 Re: Combining equations; exponents and fractions perfect_world, Thanks for the help, but I guess I'm wondering how the author arrived at equation (3), or if that is even algebraically possible. I don't think it would matter, but the context of the publication dictates that q<1 (q is porosity, which is never equal to or greater than 1, and geologically is always much lower than that). If anyone knows how to arrive at equation (3) (if it's possible), I'd be most grateful.
 August 7th, 2013, 01:34 PM #8 Senior Member   Joined: Jul 2013 From: United Kingdom Posts: 470 Thanks: 40 Re: Combining equations; exponents and fractions why would you want to get to 3? what would be the purpose of it? you could probably get there (or maybe not). weren't you given both values of a from the start?
 August 7th, 2013, 03:40 PM #9 Newbie   Joined: Aug 2013 Posts: 7 Thanks: 0 Re: Combining equations; exponents and fractions Basically, equations 1-3 are part of a whole system of equations that describe a particular concept I'm studying. I now realize that equations (3) and (4) would yield the same numerical answer, but I still don't see how (3) is obtained algebraically. If I can figure out how to arrive at (3), it will make it easier for me to understand related mathematical relationships given in other publications I'm reading.
 August 7th, 2013, 10:12 PM #10 Senior Member     Joined: Jul 2012 From: DFW Area Posts: 633 Thanks: 94 Math Focus: Electrical Engineering Applications Re: Combining equations; exponents and fractions Hi oshmunnies, Here is how I worked it: $r^{1-q}w^q=\frac{D^2}{d^2} \cdot c-\frac{D^2-d^2}{d^2} \cdot p$ $r^{1-q}=w^{-q} \left(\frac{D^2}{d^2} \cdot c-\frac{D^2-d^2}{d^2} \cdot p \right)$ $r^{1-q}=\frac{w^{-q+1}}{w} \left(\frac{D^2}{d^2} \cdot c-\frac{D^2-d^2}{d^2} \cdot p \right)$ $r^{1-q}=w^{-q+1} \left(\frac{D^2}{d^2} \cdot \frac{c}{w}-\frac{D^2-d^2}{d^2} \cdot \frac{p}{w} \right)$ $\left(r^{1-q}\right)^{\frac{1}{1-q}}=\left(w^{-q+1} \left(\frac{D^2}{d^2} \cdot \frac{c}{w}-\frac{D^2-d^2}{d^2} \cdot \frac{p}{w} \right)\right)^{\frac{1}{1-q}}=\left(w^{-q+1}\right)^{\frac{1}{1-q}} \ \left(\frac{D^2}{d^2} \cdot \frac{c}{w}-\frac{D^2-d^2}{d^2} \cdot \frac{p}{w} \right)^{\frac{1}{1-q}}$ $r^{\frac{1-q}{1-q}}=w^{\frac{1-q}{1-q}} \ \left(\frac{D^2}{d^2} \cdot \frac{c}{w}-\frac{D^2-d^2}{d^2} \cdot \frac{p}{w} \right)^{\frac{1}{1-q}}$ $r=w \ \left(\frac{D^2}{d^2} \cdot \frac{c}{w}-\frac{D^2-d^2}{d^2} \cdot \frac{p}{w} \right)^{\frac{1}{1-q}}$

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