Do these Scripture say that prayer along with fasting is more powerful than either alone? I am aware that Jesus placed much emphasis on both prayer and on fasting. And I am not really sure that the following Scriptures call for doing both at the same time, but it can be taken to believe that it is referring to simultaneous prayer and fasting.
All Scriptures are from the King James translation

Matthew 17:21  Howbeit this kind goeth not out but by prayer and fasting.
Mark 9:29  And he said unto them, This kind can come forth by nothing, but by prayer and fasting.

Would someone more versed in Biblical interpretation give me some guidance.
Let's break down the scriptures you used.

This kind...

In Matthew 12:45 we read that there is a demonic hierarchy. Some demons are more wicked, or more powerful, than others.


  Matthew 12:45 Then goeth he, and taketh with himself seven other
  spirits more wicked than himself, and they enter in and dwell there:
  and the last state of that man is worse than the first. Even so shall
  it be also unto this wicked generation.


In the Old Testament as Daniel was visited by an angel he was told that he needed help from Michael to defeat a greater power so that he could bring the revelation to Daniel.


  Daniel 10:12-13 Then said he unto me, Fear not, Daniel: for from the
  first day that thou didst set thine heart to understand, and to
  chasten thyself before thy God, thy words were heard, and I am come
  for thy words. But the prince of the kingdom of Persia withstood me
  one and twenty days: but, lo, Michael, one of the chief princes, came
  to help me; and I remained there with the kings of Persia.


Prayer elevates the worshiper into the supernatural and causes the things of this world to fade away. It dims the eye to the worries and cares of the natural world. Prayer is so important that in Acts the disciples appointed the first deacons to administer to the widows and so forth so that they could be more devoted to prayer (and study). It is far easier to minister when you have spent time in His Presence. 


  Acts 6:4 But we will give ourselves continually to prayer, and to the
  ministry of the word.


Fasting, likewise, denies the things of this world to better serve him.

On fasting, it's important to note that when Jesus fasted forty days he was not hungry until after the fast was over.

Luke 4:2
Being forty days tempted of the devil. And in those days he did eat nothing: and when they were ended, he afterward hungered. 



So this kind of spirit would not come out except by the faith of someone who had yielded themselves to God through prayer and fasting. No doubt the disciples had spent the day in fellowship with other disciples, but had neglected to spend time with their heavenly Father. When a test came, their (spiritual) eyes had not been sufficiently dimmed to the things of the world and they could not cast out the evil spirit.

Should you fast and pray simultaneously? Here is some advice based on my personal experience. I would encourage you to try praying early in the morning before breakfast, or skip the last meal in the day and pray into the evening. In this way you can both fast and pray as I believe the early believers did. If you feel led to fast further, and do not find yourself growing hungry but instead are being fed by fellowship with the Lord, continue fasting. In my experience fasting is easiest when I have something or someone weighing on my soul and need an answer from God desperately.