# Application of fractional part in expansion

**Algebra**Level 5

\(\displaystyle (a+b\sqrt c)^{n}= I+f\) where \(\displaystyle I\) is integer part and \(\displaystyle f\) is fractional part..

Which of the following are true if \(\displaystyle a^{2}-1=b^{2}c\)

Value of \(\displaystyle 1-f\) is \(\displaystyle I\)

Value of \(\displaystyle 1-f\) is \(\displaystyle \frac { 1 }{ I } \)

Expansion of \(\displaystyle (a-b\sqrt c)^{n}\) is \(\displaystyle 1-f\).

Value of \(\displaystyle 1+f\) is \(\displaystyle I\)

Value of \(\displaystyle 1+f\) is \(\displaystyle \frac { 1 }{ I } \)

Expansion of \(\displaystyle (a-b\sqrt c)^{n}\) is \(\displaystyle 1+f\).

\(\displaystyle \frac{1}{1-f}-f=I\)

\(\displaystyle \frac{1}{1+f}-f=I\)

Submit the answer as the product of the true statements.

For example if 2,3,5 are true submit the answer as 30

\(\displaystyle 0 <a-b\sqrt c<1\) , n is a positive integer. a>1, c>0