I am not certain if this could be answered in any definitive manner.  I think a case can be made that each branch in its own right is extremely powerful.  I would say that the power to determine constitutionality is a powerful element and one that rests solely in the power of the judicial branch.  The fact that no other branch can determine this is what makes, in my mind, the judicial branch so powerful.  With one determination, acts of Congress and actions of the President can be nullified in the assertion of not following the letter or spirit of the Constitution.  I would constitute this as power, and something that only the Judicial branch possesses.  Yet, I would say that a case can be made for all of the branches for being "most powerful," the essence of what the Framers' must have intended.

A matter of opinion, I suppose.  I could argue the Executive Branch because the power is concentrated in the hands on one person, and they control the military.  I could argue the Legislative Branch, since they have the power of the purse - the power to write the checks which would fund the military or anything else.  I could also say the Supreme Court, since it has the final say on all challenged laws, there is no higher word, their appointments are lifetime and they are insulated from elections.

So take your pick! :-)

You can make an argument for any of them, but I'm going to go with the Presidency or executive branch.  Here's why:

Congress has a lot of power.  In fact, it has all the power to make laws.  So you would think they have the most power.

But in our current system, with the power of the media and all, the President has the most power.  He is the one person that just about everyone in the country recognizes.  When he talks, the TV stations cover his speeches and pay attention to what he says.

This means that the President, more than anyone else, can decide what issues are going to be on the agenda.  In addition, he has more of an ability than anyone else to get people to pay attention and listen to his point of view.

What is the most powerful branch of goverment?i need to write a report

In the end, it might not be as important to identify the most powerful branch of government as much as it is to recognize the interdependence of all the branches in order for government to properly function.  The framers were wise enough to understand that a government which is dependent on its multiple branches to ensure a smooth and orderly administration of government is probably going to be more successful and endure longer than one which places primacy on one function of government over another.  A plausible case can be made for all three branches having separate powers that might be deemed superior and more powerful than the other two, but the reality is that within the principle of checks and balances, each branch has its own power that is limited by the other two.  This creates a triangulation of power where effective government is realized.

What is the most powerful branch of goverment?i need to write a report

This question has been asked recently, and is mostly a matter of opinion, but I tend to think the Executive Branch is.  The President is the Commander in Chief of the military, and may use signing statements and executive orders to legislate some things just as Congress does.  He is able to nominate all new members of the Supreme Court, and the branch is concentrated in just one person, as opposed to divided like the other two branches.

Included below are some links to previous answers.

what is the most powerful branch of government?  

All three branches of government in the U.S.A., the Executive, Legislative and Judicial, were made to be equal. One branch does not over power the other. This was one of the main concerns of the Founding Fathers when they were creating the Constitution. This is why the systems of checks and balances, and the separation of powers (idea originally by Montesquieu) are used. The three branches and their powers can be found in the first three articles of the Constitution. The people give the power to the government, as it says "We the People," in the Preamble to the Constitution, and this power is divided to keep balance between order and liberty. All branches are equally important and serve different purposes.

A good book and easy read that would help you to understand the branches more in detail is The U.S. Constitution for Everyone by Jerome Agel and Mort Gerberg, a Perigee Book.