# Homework Help: Question regarding binomial theorem.

1. Jul 11, 2012

### sankalpmittal

1. The problem statement, all variables and given/known data

(√2 + 1)6 = I + f

Where I is the sum of integer part of the expansion of (√2 + 1)6 and f is sum of the fraction part in (√2 + 1)6.

2. Relevant equations

(x+1)n = nC0 xn + nC1 xn-1 + nC2 xn-2 + ....... + nCn

nCn = nC0 = 1

3. The attempt at a solution

I expanded (√2 + 1)6 , then simplified and then got the expression 44+99√2/2.
Then I got I=44 which was not even the correct answer. The correct was 197. This question is a competitive level question.

2. Jul 11, 2012

### Infinitum

Hi sankalp

The answer you get is incomplete. The second term also has an integral and fractional part itself. You need to add them to get your answer

Another way to see it is that f is defined to be between 0 and 1, so is your fractional part between zero and one?

PS : Your expansion itself seems incorrect to me. Recheck it.

3. Jul 12, 2012

### sankalpmittal

Hii Infinitum!!

One way to do is to seriously find (√2 + 1)6. But that will be a noob way.

Ok , so on expanding , I get :

(√2 + 1)6 = 6C0 8 + 6C1 4√2 + 6C2 4 + 6C32√2 + 6C4 2 + 6C5 √2 + 6C6

Now on solving , I get (√2 + 1)6 = 99+ 70√2
I + f = 99+ 70√2

Now what else can I do ? Any hint ?

4. Jul 12, 2012

### Infinitum

Yep. That sounds correct. Now you can use the approximate value of √2 to multiply, and hence get the integral part of the expansion.

Hint : You only need to use 1.41 as your approximation, as any more digits will not change effect the integral part

5. Jul 12, 2012

### sankalpmittal

Did not recognize that this was so simple...

99+ 70√2

99+ 70(1.41)
I = 99+98 = 197 !!

Awesome!!

Edit : All right , thanks for the efforts.....

Last edited: Jul 12, 2012
6. Jul 12, 2012

### Infinitum

The method your professor uses can be applied in general to all such problems, so it is good in its own right. The one I suggest requires that you know the value of √2, which is frequently used and known. What about when it is √327? You would have to first find the square root, then approximate.