1. ## solving for unknowns

1.) If,
{4x^2 - sqrt[3]{x^2} + 3}/{sqrt[6]{x^5}} = 4x^a - x^b + 3x^c}

then a= ______ , b= _______ , and c= _______ .

2.) If V=4r^2h and r/(H-h)= R/H, write V as a function of R
V= _______________

3) Solve for
S= (a^n) -1 /i

a= __________

4.) Solve for n:
P= R[(1-x^-n)/i]
n=___________

2. I'll give you a headstart on the first one ...
$\displaystyle \frac {4x^2 - \sqrt[3]{x^2} + 3}{\sqrt[6]{x^5}} = 4x^a - x^b + 3x^c$

Multiply through by that $\displaystyle \sqrt[6]{x^5}$ for a start, and use fractional indices to make it easier to read:

$\displaystyle 4x^2 - x^{2/3} + 3 = x^{5/6}\left({4x^a - x^b + 3x^c}\right)$

Then equate the terms with coefficients 4, -1 and 3 and remember that when multiplying two terms you add their indices.

3. 1.) If,
{4x^2 - sqrt[3]{x^2} + 3}/{sqrt[6]{x^5}} = 4x^a - x^b + 3x^c}
then a= ______ , b= _______ , and c= _______ .

Does sqrt[3]{x^2] mean cuberoot of x^2? And sqrt[6](x^5) the 6th root of x^5?
If yes, then,
[4x^2 -x^(2/3) +3] / [x^(5/6] = 4x^a -x^b +3x^c
4x^(2 -5/6) -x^(2/3 -5/6) +3x^(-5/6) = 4x^a -x^b +3x^c
4x^(7/6) -x^(-1/6) +3x^(-5/6) = 4x^a -x^b +3x^c

Therefore, a = 7/6; b = -1/6; and c = -5/6 ------------answer.

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2.) If V=4r^2h and r/(H-h)= R/H, write V as a function of R
V= _______________

The idea here is to put V and R into one equation only.

We see that h is common in the two given equations, so we work on that:
From 1st equation, h = V/(4r^2).
Substitute that into the 2nd equation,
r /[H -V/(4r^2)] = R/H.
Cross multiply,
rH = R[H -V/(4r^2)]
rH = RH -VR/(4r^2)
Isolate V,
VR/(4r^2) = RH -rH
V = (RH -rH) /[R/(4r^2)]
V = (4r^2)(RH -rH) /R

Or, V = (4Hr^2)(1 -r/R)
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3) Solve for
S= (a^n) -1 /i

a= __________

The idea here is to just isolate a.

iS = a^n -1
a^n = iS +1
Take the log of both sides,
n*log(a) = log(iS +1)
n = log(is +1) / log(a) -------------answer.

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4.) Solve for n:
P= R[(1-x^-n)/i]
n=___________

Just isolate n:

P/R = (1 -x^(-n) / i
iP/R = 1 -x^(-n)
x^(-n) = 1 -iP/R
Take the log of both sides,
(-n)log(x) = log(1 -iP/R)
-n = log(1 -iP/R) / log(x)
n = -log(1 -iP/R) / log(x) -------------answer.

Can also be:
n = -log[(R -iP)/R] / log(x)
n = [-log(R -iP) +log(R)] / log(x)
n = [log(R) -log(R -iP)] / log(x) ------answer also.