Question

Let x = (5√2 + 7)¹⁹, then x{x} ({.} denotes the fractional part of x) is equal to

(A) 2¹⁹

(B) 3¹⁹

(C) 0

(D) 1

Answer 1

Correct option is (D) 1

x = (5√2 + 7)¹⁹

= I + f

= [x] + {x}    ....(1)

We see that, 

0 < 5√2 - 7 < 1

0 < (5√2 - 7)¹⁹ < 1

Thus, (5√2 - 7)¹⁹ = fractional part

= {x}   ....(2)

Multiplying (1) and (2),

x{x} = (5√2 + 7)¹⁹ (5√2 - 7)¹⁹ 

= [(5√2 + 7) (5√2 -7 )]¹⁹ 

= [(5√2)² - 7²]¹⁹ 

= [50 - 49]¹⁹ 

= 1¹⁹ 

= 1