#1 - I am not going to tell you.  I want you to figure this one out.  I will start you off though.

(9^x)/(3^x) = (9/3)^x

27 = 3^3

I think I might have even told you too much but I will let you figure this one out from here.

#3 - The answer IS 7.

We know an acute triangle has angles less than 90 degrees and an obtuse triangle has an angle more than 90 degrees.  We can use this information for this problem.

If a triangle has a 90 degree angle the following is true:

a^2 + b^2 = c^2

So if we wanted to solve for c (or the side we are looking for), we simply use:

sqrt(a^2 + b^2) = c

BUT, if we want to guarantee that this was an acute triangle then the other side would have to be less than c, and for an obtuse triangle the other side would have to be greater than c.

So we can make the following rules if given 2 sides of a triangle and looking for the third:

For an acute triangle the third side 'c' would have to satisfy:

c < sqrt (a^2 + b^2)

For an obtuse triangle the third side 'c' would have to satisfy:

c > sqrt (a^2 + b^2)

Note:  If the sqrt(a^2 + b^2) is an integer, then the largest value such that it is an acute triangle would be one less than that number.  Alternately, for an obtuse triangle then the smallest value for c would be one more than that number.

Ex [1]  An acute triangle has integer sides of 3, x, and 7 units. the largest value of x is __________.

1. 
   1st determine sqrt (3^2 + 7^2) = sqrt(58);


2. 
   The sqrt(58) is between the number 7 and 8.


3. 
   The smallest value x can be is 7.


4. 
   The answer is 7.


5. 
   If the question had asked for the smallest value of x to make it into an obtuse triangle, the value would be 8.

Ex [2]  An obtuse triangle has integral sides of 5, x, and 12.  The smallest value of x is __________.

1. sqrt(5^2 + 12^2) = 13.  *You should recognize the pythagorean triple.


2. The smallest value is one more than 13 or 14.


3. The answer is 14.


4. If the question had asked for the largest value of x such that it makes an acute triangle the answer would be 12.

I hope I made myself clear. 

Last Question:

 I believe most people will tell you just to change to base 10 and then change back to base 4, but that is not how I would do it.  I only do that when I am dividing in a different base, not when I am multiplying.

First treat this like you would normally treat it in base 10.  If I saw 22 x 33 I would change this to be 11 x 66.  Now the problem is easy.  So do the same thing here. 

Now the tricky part.  You do the same thing you normally would do except change everything to base 4.  Let me show you the steps:

1. Usually, we would write 6.  But in base 4 there are no 6's so we need to change 6 to base 4.  It is 12.  Write 2, carry *1.


2. 6+6 = 12 + *1 = 13.  In base 4, 13 is 31.  So write 1, carry *3.


3. 6+*3 = 9.  In base 4, 9 is 21.  Write 21.


4. The answer is 2112.

Let's look at another example:

Ex [1]  (32 base 5) x (12 base 5) = _________(base 5).

1. I am assuming you would multiply the same way I would which is found in the Multiplying Numbers section under Multiplying By 12-19.


2. First, we multiply 2 x 2 = 4.  4 in base 5 is 4.  Write 4.


3. 2*3 = 6 which is 11 in base 5.  32 (base 5) + 11 (base 5)= 43 (base 5).  Write 43.


4. The answer is 434.

This style of thinking is more difficult than just multiplying by a single digit, but as you can see it can be done the same way.  If this is difficult to see, I would suggest practicing on them.  If you still can't seem to get it, then I would change to base 10 then change back to base b.

Webmaster

P.S.  I don't mind being called Mr. Jones.  I prefer it to the informal, impersonal "webmaster".  Although I will usually have a signature of webmaster.

Actually, I had that same question.  Usually when doing these problems, the question will ask it such that the last value is c. 

But my reasoning was this.  I am looking for the largest possible value for x.  That means if I treat x as a leg I will not get as high a number as if  treated it as the hypotenuse.

Just because the problems usually state the legs in order does not mean they have to.  After thinking on it I might be wrong about the obtuse triangle, but I am pretty sure about the acute triangle.  If it is acute, always treat the missing value as the hypotenuse.  I will look to see about the obtuse triangle.  Very good question.

As for your other question, I am currently in Beijing.  We (my wife and I) have lived here for almost 2 years now, thinking about staying for a 3rd.  Of all things, I am an English teacher here.  Since I can only understand a little Chinese (not enough to follow the television) I have to find other ways of using my time.  So I thought I would work on a website.  Are you really Chinese?  Do you understand Chinese, because from the way you write, you have EXCELLENT English skills, which is unusual for a native Chinese.

Webmaster

Ni hao. 

How do you say "Heck Yeah" in Chinese?  If you come for the summer you can definitely look us up.  We are teachers at "Hua gong da xue" or the Beijing University of Chemical Technology.  We are hopefully going to be here for the summer and we would love (my wife especially) to get some real Chinese lessons.  And you already know I love number sense.  I would love to be your teacher.  How old are you anyway and what school are you with? 

If you do come, I might even be able to write up some sample tests for you if I have enough time.

Zai jian,

Webmaster

P.S.  Does your family know English?  Maybe I can look them up or give them a call.