(9x^3y^4)/(16a^4b^2) divided by (45x^4y^2)/(14a^7b) 

  9x3y4     45x4y2
-------- ÷ --------
 16a4b2      14a7b

Invert the second fraction and change 
division to multiplication:

  9x3y4      14a7b 
-------- × --------
 16a4b2     45x4y2

Indicate the multiplication of the numerators
and the denominators so that you have just one
fraction:

  (9)(14)x3y4a7b 
------------------
 (16)(45)a4b2x4y2

Begin by simplifying the numbers.  Cancel the
9 into the 45, and divide the 14 and the 16
each by 2:

   1   7
  (9)(14)x3y4a7b 
------------------
 (16)(45)a4b2x4y2
  8   5

So now we have:

   7x3y4a7b 
 ------------
  40a4b2x4y2

There are no eponentials with like 
bases in the numerator or in the 
denominator to add the exponents of.
So we subtract exponents of 
exponentials with like bases
between numerator and denominator:

We see there is an x3 on top and
an x4 on the bottom. So we subtract
larger minus smaller 4-3 and get 1
and put x1 on the bottom because
the larger exponent, 4, was on the
bottom, and eliminate x from the
top.

    7y4a7b 
 ------------
  40a4b2x1y2

We see there is a y4 on top and
a y2 on the bottom. So we subtract
larger minus smaller 4-2 and get 2
and put y2 on the top because
the larger exponent, 4, was on the
top, and eliminate y from the
bottom.

   7y2a7b 
 ----------
  40a4b2x1

We see there is an a7 on top and
an a4 on the bottom. So we subtract
larger minus smaller 7-4 and get 3
and put a3 on the top because
the larger exponent, 7, was on the
top, and eliminate a from the
bottom.

   7y2a3b 
  --------
   40b2x1

We give the b on top its understood
exponent of 1

   7y2a3b1 
  ---------
    40b2x1

We see there is a b1 on top and
a b2 on the bottom. So we subtract
larger minus smaller 2-1 and get 1
and put b1 on the bottom because
the larger exponent, 2, was on the
bottom, and eliminate b from the
top.

    7y2a3 
   --------
    40b1x1

We are done except for erasing the
1 exponents, leaving them understood:

    7y2a3 
   -------
    40bx

That's the final simplified answer.

Edwin