Question

without actually performing long division state whether the rational number will have a terminating decimal expansion or a non decimal expansion:47/2⁴×5²​

Answer 1

Answer:

The rational number 47/2^4×5^2 would have a terminating decimal expansion.

Step-by-step explanation:

$If \: the \: denominator \: of \: the \\ \: rational \: number \: could \: be \: \\ expressed \: as \: 2 {}^{m} \times 5 {}^{n} ; where \: m \: \\ and \: n \: are \: rational \: numbers, \\ the \: rational \: number \: is \: \\ convertible \: into \: a \: terminating \: \\ decimal. \:$

$Since, \: 2 {}^{4} \times 5 {}^{2} \: \: are \: already \\ \: expressed\:as \: 2 {}^{m} \times 5 {}^{n} . \\ Therefore, rational \: number \\ \: \frac{47}{2 {}^{4} \times 5 {}^{2} } \: is \: a \: terminating \\ \: decimal .\:$

OR

Also, We knew that the

terminating decimal is a decimal that ends with finite number of digits.

=>47/2^4×5^2 = 47/400

= 0.1175

The resulting decimal number ends with four decimal digits and hence, 47/2^4×5^2 has a terminating decimal expansion.

( Here, (^) represents => raise to the power of . )*