Secret formulas "Fast" Mathematics

Once, when I was new to the new student's first level, I became acquainted with one of the other new students who later became my best friend. When the initial introductions, we were chatting here and there. Ask questions here and there. Answer it, answer here. Until he was finally told bahwaa Basic Mathematics test scores of his, that is one of the subjects tested in UMPTN *, is 100 alias correctly all.
Hearing his story, I was amazed he made. In my mind, I concluded "Well he certainly was a very clever". My admiration encouraging my curiosity about his knowledge in mathematics. Finally, in the early days of introduction, I invite he talked about the mathematics that we've learned when during the elementary through high school.From the conversation, I learned, it turns out he really broad knowledge of mathematics that have been learned. Until finally, perhaps to show his skill, he invited me to shoot quickly solving math problems.Received the challenge, I actually ngeper too. Since I was not as smart as him. However, because this name is also not a race and it's nothing, I still would have the time. The questions were selected at random from the book of practice questions tests and Exams UMPTN * ** a few years earlier that he was still diligently carry everywhere. Then, the race quickly solve math problems began.How did it go? Who's fastest?Sure enough, within minutes, my friend managed to solve all the problems we had chosen earlier (for the chosen only 3 problems still.) And he came out as the fastest, the winner. While I, one question I was not able to finish. At that time, I too struggled with the issue the first number is quite difficult for my size at that time. Despite the best of my ability try to finish it, but it turns out, until time runs out has not been met as well. I also acknowledge the advantages and prowess.With a little sheepishly, I asked him about a matter which I could not finish it. While I also ask why he so quickly can solve these problems. About the time that I can not finish is as follows.Problem: When a 1 / a = 5, then the value of a3 1/a3 = ...With a fast friend of mine that was solving the problem are as follows:1/a3 a3 = (a 1 / a) 3 - 3a.1 / a (a 1 / a) = 53-3 (5) = 125-15 = 110.Seeing part of the solution, I can only stare at that time. "Just one line? Though I tried to finish it rows and rows, and have not met too ", it is in my mind. Then, I asked to my friend, why the way the process like that?With pleasure, he also explained to me. He said that, these kind of problems can be easily solved with the formula "quickly" following.a3 b3 = (ab) 3 - 3ab (ab) ... ... ... ... ... ... ... ... ... ... ... ... .. (1)By replacing b by 1 / a, he said, the matter was resolved quickly as she had done earlier.I am not accustomed to using the formula of "fast" when in high school, curious to know the reason why the formula "quickly" can be used. But unfortunately, my friend did not tell me. Instead he quickly adds another formula that had he ketahuinya, namely:a3 - b3 = (a - b) 3 3ab (a - b) ... ... ... ... ... ... ... ... ... ... ... .... (2)Finally, all the chat was wrong. He hurried back to his boarding-boarding. I was so, go home with a sense of curiosity is up.In kost-kost-an, with a sense of curious to know, I was tinkering with the formula "quickly" that had he used it. After some time, finally, also solved the secret formula "quick" my friend used it. I managed to trace the origin of the formula "quickly" is, managed to uncover his secret. (Duh seemed so happy, I can not express with words.)The results of my search, after I trim, such as the following.(A b) 3 = (a b) 2 (a, b)= (A2 2ab b2) (a b)= A3 a2b b2a b3 2a2b 2ab2= A3 b3 3a2b 3ab2A3 b3 = 3ab (a, b)So, (a b) 3 = a3 b3 3ab (a, b).So, a3 b3 = (a b) 3 - 3ab (a, b). The formula "quickly" (1) I can prove its truth. Then, in a similar way, I also managed to trace the origin of the formula "quickly" (2).Although what I have done is simple, but for my size when it is something joyous heart, pleasant thoughts, and quench the thirst of my curiosity.Since that time, if there are formulas "rapid" that I met in the books guidance tests, I was encouraged to trace its origin. That way, I often managed to solve the secret formula "quickly" that had been widely circulated among the students who follow the guidance test.Well, that much was just the story ... yes, next time, God willing, I will discuss the merits of the use of formula "quickly" (There is one very funny story about it. Want to know? Please wait in the next post ...). That's it ... yes, hopefully useful.As training material for you, try searching the origin of the formulas "rapid" below.

   1. Equation of the line through the point (0, a) and (b, 0) is the ax by = ab.
   2. Consider the following picture. The length of PQ can be determined easily, namely:
      PQ = (AP. DC DP. AB) / (AD)

Note:* UMPTN: Entrance Exam State University (now called SNCA)** Exams: Learning Evaluation Final Stage of the National (now called UAN)
source: mathematicse.wordpress.com