SOLUTION: 1. IF X=7-4RT3, FIND THE VALUE OF RTX+1/RTX? 2. X=3+2RT2, FIND THE VALUE OF X^3-1/X^3?

Algebra ->  Human-and-algebraic-language -> SOLUTION: 1. IF X=7-4RT3, FIND THE VALUE OF RTX+1/RTX? 2. X=3+2RT2, FIND THE VALUE OF X^3-1/X^3?       Log On


   

Question 983439: 1. IF X=7-4RT3, FIND THE VALUE OF RTX+1/RTX?
2. X=3+2RT2, FIND THE VALUE OF X^3-1/X^3?
Answer by ikleyn(52798) About Me  (Show Source):
You can put this solution on YOUR website!

1. Let me re-formulate your first request in the correct algebraic form:

      If  x = 7-4sqrt%283%29,  find the value   sqrt%28x%29 + 1%2Fsqrt%28x%29.

Solution

We have  x = 7-4sqrt%283%29.  Let  y be  7%2B4sqrt%283%29:  y = 7%2B4sqrt%283%29.

Then  x%2Ay = %287-4sqrt%283%29%29.%287%2B4sqrt%283%29%29 = 7%5E2 - %284sqrt%283%29%29%5E2 = 49 - 16%2A3 = 49 - 48 = 1.     (I applied here the formula %28a-b%29%2A%28a%2Bb%29 = a%5E2+-+b%5E2.   Are you familiar with it ?)

Thus we obtain a remarkable formula: %287-4sqrt%283%29%29.%287%2B4sqrt%283%29%29 = 1.

It means that the numbers  7-4sqrt%283%29  and  7%2B4sqrt%283%29  are reciprocal each to the other:  their product is equal to  1.

In other words,  x = 1%2Fy,   or   y = 1%2Fx,     i.e.   1%2Fx = 7%2B4sqrt%283%29.

Now,  let us square   sqrt%28x%29+%2B+1%2Fsqrt%28x%29:

%28sqrt%28x%29+%2B+1%2Fsqrt%28x%29%29%5E2 = %28sqrt%28x%29%29%5E2 + 2%2Asqrt%28x%29%2A%281%2Fsqrt%28x%29%29 + 1%2F%28sqrt%28x%29%29%5E2 = x + 2 + 1%2Fx.     (I applied here the formula %28a%2Bb%29%5E2 = a%5E2+%2B2ab+%2B+b%5E2.   Hope you are familiar with it.)

Substitute here   x = 7-4sqrt%283%29  and   1%2Fx = 7%2B4sqrt%283%29  as we found above.

You can continue this identity as

%28sqrt%28x%29+%2B+1%2Fsqrt%28x%29%29%5E2 = %28sqrt%28x%29%29%5E2 + 2%2Asqrt%28x%29%2A%281%2Fsqrt%28x%29%29 + 1%2F%28sqrt%28x%29%29%5E2 = x + 2 + 1%2Fx = 7-4sqrt%283%29 + 2 + 7%2B4sqrt%283%29 =   (now collect the common terms and annihilate the terms with  sqrt%283%29 ):

= 7 + 7 + 2 = 16.

Since   %28sqrt%28x%29+%2B+1%2Fsqrt%28x%29%29%5E2 = 16,  you get   sqrt%28x%29+%2B+1%2Fsqrt%28x%29 = 4.

Amazingly,  isn't?

The solution is completed.

The problem and the solution are like sweet cookies.  Thank you  (and yours teacher)  for giving it.


2.  Regarding the second problem of your request,  try to solve it yourself similar to the first one.

You have   x = 3%2B2sqrt%282%29.

Introduce   y = 3-2sqrt%282%29.

Check that   x%2Ay = 1,   i.e. the numbers   3%2B2sqrt%282%29  and   3-2sqrt%282%29  are reciprocal   and   1%2Fx = 3-2sqrt%282%29.

Based on it,  complete the solution yourself.

Good luck !!!