Question

Without actually performing the long division state whether the following rational numbers will have terminating decimal expansion or non terminating repeating decimal

WIth steps

19/250 , 23/2048 ,127/910 ,35/800 / 54/434 , 52/2³*5^4

Answer 1

terminating number's denominator is only in the form 2^n*5^m  where m and n are real number

∴

(1)  19/250=19/(2^1*5^3)     where m=3 and n=1     hence terminating

(2) 23/2048=23/(2^11*5^0)     where m=0 and n=11     hence terminating

(3) 127/910=127/(2^1*5^1*7^1)    where m=1 and n=1   having factor 7   hence non-terminating

(4)  54/434=54/(2^1*5^0*7^1*13^1)    where m=0 and n=1    having factor 7 and 13  hence non-terminating

(5)  52/(2^3*5^4)   where m=4 and n=3     hence terminating

Answer 2

(a) 19/250 = 19/2x5³

As the denominator is in the form 2^n x 5^m.

It will have terminating decimal expansion.

(b)23/2048= 23/2^11

As the denominator is in the form 2^n x 5^m.

It will have terminating decimal expansion.

(c) 127/910 = 127/2 x 5 x 7 x 13

As the denominator is not in the form 2^n x 5^m.

It will have non-terminating decimal expansion.

(d) 54/243 = 3³ x 2/3^5

As the denominator is not in the form 2^n x 5^m.

It will have non-terminating decimal expansion.

:)