Expert: Sherman D. - 4/12/2005

Question
-------------------------

Question -
!simplify the following questions. When possible reduce to the simplest form and evaluate. thank you ever so much for your help!!
^ = exponent
a)
(a-b_^3 .(x+y+^2  .(3p-9)^4
--------------------------------
3(x+y) .(3p-9)^3 .(a-b)^3


  
b)

  5z^-3
  -------
  15z^-5


c)

  16^1/4


d)

  if logb  8=3 then b=-----------


e)

  if log4 64=x, then x =------------

f)

if f(x) =cos3x, then f(pie symb/6) is-------------

g)   csc^4x-2csc^2 xcot^2 x+cot^4x

Answer
Let me see if i get this straight

a)
((a - b)^3 * (x + y)^2 * (3p - 9)^4) /
(3(x + y) * (3p - 9)^3 * (a - b)^3

((a - b)^3 * (x + y)^2 * (3p - 9)^4) /
((a - b)^3 * 3(x + y) * (3p - 9)^3

as you can see (a - b)^3 cancel out, leaving you with

((x + y)^2 * (3p - 9)^4) / (3(x + y) * (3p - 9)^3)

but it doesn't stop there

(((x + y)^2)/(3(x + y))) * (((3p - 9)^4)/((3p - 9)^3)

(1/3)(x + y)^(2 - 1) * (3p - 9)^(4 - 3)

(1/3)(x + y) * (3p - 9)

or

((x + y)(3p - 9))/3

(3(x + y)(p - 3))/3

answer is

(x + y)(p - 3)

-----------------------------------------------------------

b)
(5z^(-3))/(15z^(-5))

(5/15) * z^(-3 - (-5))
(1/3) * z^(-3 + 5)
(1/3) * z^2
(z^2)/3

-----------------------------------------------------------

c)
16^1/4
16^(1/4) = 2

-----------------------------------------------------------

d)
if log(b)8 = 3 then b =

3^b = 8
(log(8))/(log(3)) = b
b = 1.8928

-----------------------------------------------------------

e)
if log(4)64 = x

4^x = 64
x = 3

or

x = (log(64))/(log(4))
x = 3

-----------------------------------------------------------

f)

if f(x) = cos(3x), then f(pi/6)

pi/6 = 180/6 = 30

f(pi/6) = cos(3x)
becomes
f(30) = cos(3(30))
f(30) = cos(90)
f(30) = 0

so

f(pi/6) = 0

-----------------------------------------------------------

g)
csc(x)^4 - 2(cscx^2)(cotx^2) + cot(x)^4

(1 + cotx^2)^2 - 2(1 + cotx^2)(cotx^2) + cot(x)^4
(1 + cotx^2)^2 - 2(cotx^2 + cotx^4) + cot(x)^4
(1 + cotx^2)^2 - 2cotx^2 + 2cotx^4 + cot(x)^4
((1 + cotx^2)(1 + cotx^2)) - 2cotx^2 + 3cotx^4
(1 + cotx^2 + cotx^2 + cotx^4) - 2cotx^2 + 3cotx^4
1 + 2cotx^2 + cotx^4 - 2cotx^2 + 3cotx^4
1 + 4cotx^4

That is what i got, however when i did it another way, i got 1, but i am not completely certain if 1 is the answer. If you like, you can check behind my work to make sure i didn't do anything wrong.

cscx^2 = 1 + cotx^2 found at www.alcyone.com/max/reference/maths/trigonometry.html