Are these Calculus problems done correctly?

Question

So yeah, I have a 46 question Calculus packet due the first day of school. I finished a while ago but now I'm checking my work since its going to be graded.

I came across a few I'm not sure on, so I figured I would ask here and some other places to see if anyone can help.

21. Rewrite with fractional exponents:

a) sqrt(1+x^2)

B) 1/(sqrt(1+z^2)^3)

For a, I answered (1+x^2)^(1/2) but something tells me I can simplify it. At the same time, I checked some stuff on my calculator and it doesn't look like I can, but I'm not sure.

For b, I answered (1+z^2)^-3 - same thing here, just not sure.

25. A piece of wire 5 inches long is to be cut into two pieces. One piece is x inches long and is to be bent into the shape of a square. The other piece is to be bent into the shape of a circle. Find an expression for the total area made up by the square and the circle as a function of x.

Okay, this one was kind of hard for me, but this is what I came up with (not sure if this is right):

a_s = s^2

a_s = (x/4)^2

a_s = (x^2)/16

c = 5-x

c = 2(pi)r

5-x = 2(pi)r

(5-x)/2pi = r

A_c = (pi)(r^2)

A_c = (pi)((5-x/2pi)((5-x)/2pi))

A_c = (pi)(-x+5)^2/4(pi)^2

A_final = x^2/16 + pi(-x+5)^2/4(pi)^2

A_final = (pi^2 * x^2)/16pi^2 + 4pi(-x+5)^2/16(pi)^2

A_final = (pi)(x^2)+4(x^2-10x+25)/16pi

A_final = (pi)x^2 + 4x^2 - 40x + 100/16pi

...Phew.

Okay, I think that's all.

Any help is appreciated! Thank you.

Answer 1

Your part a is correct, and you really cannot simplify it.

Your part b is almost right, except instead of it being (1+z^2)^-3 it must be (1+z^2)^-(3/2) because you are solving for fractional exponents and there was a square root in your equation.  (You might have had that and just forgot to write it; I just wanted to make sure.)

Your final problem is also correct; nice work by the way.