Alan

Two approaches below.  One uses L'Hopital and one uses a series expansion:

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"how do you simplify 32 with an exponent of 3 over 5"

\(32^{3/5}\rightarrow(32^{1/5})^3\rightarrow2^3\rightarrow8\)

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You cannot obtain an explicit algebraic solution in the form x = something here, where the 'something' is independent of x.

A possible numerical approach is to use the Newton-Raphson method, where you have to subtract one side from the other to create a function that is equal to zero:

The positive solution here is x = 2.314.  The negative solution could be obtained by using an initial guess of, say -3, though the symmetry here makes it obvious that the negative solution is x = -2.314.

You would need to use a similar numerical method for your other suggestion.

Here one obvious solution is x = pi.  Use Newton-Raphson to find the others.  For example:

So another solution is x = 0.828 (and because of the sine curve, the final solution is at x = 2pi - 0.828).

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Let x = 0.45454545...

then 100x = 45.45454545...

Subtract x Fromm 100x to get:  99x = 45

x = 45/99 or  x = 5/11

There are 3 solutions.  One is obvious from the graph below. The other two can be found numerically (one way is to guess, using the graph as a starting point, and then refine your guess until you get as close as you want).

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You have succeeded!

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Imagine a right angled triangle with opposite = 2 and hypotenuse = x

Its adjacent is then sqrt(x^2 - 2^2), so the tangent of the angle is opposite/adjacent =  2/sqrt(x^2 - 2^2)

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I'll assume v is the final velocity, and that the acceleration is constant, in which case you can use:

(i) d = u*t +(1/2)*a*t^2   where d = distance,  u = initial velocity, a = acceleration and t = time

and

(ii)  v = u + a*t  

I'll also assume the acceleration is 4 m/s^2 

(i)  1400 = 0 +(1/2)*4*t^2   so  t = sqrt(700) secs  ≈ 26.5 secs

(ii) v = 0 + 4*t   so v = 4*sqrt(700) m/s  ≈ 106 m/s

A. 

B. You should be able to use the graph above to answer part B.

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Is V the final velocity, the average velocity or what?  Is the starting velocity zero?