## Fractional part of √5 and √10

$frac \, ( \,\sqrt { \,5 \,} \,) \; = \; \sqrt { \,5 \,} \; - \; 2$

$frac \, ( \,\sqrt { \,10 \,} \,) \; = \; \sqrt { \,10 \,} \; - \; 3$

Prove that for any positive integer   $n$

(1)   there exists an integer   $A$   such that

$(\sqrt { \,5 \,} \; - \; 2)^n \; = \; \sqrt { \,A + 1 \,} \; - \; \sqrt { \,A \,}$

(2)   there exists an integer   $B$   such that

$(\sqrt { \,10 \,} \; - \; 3)^n \; = \; \sqrt { \,B + 1 \,} \; - \; \sqrt { \,B \,}$