Question

aaahhh guyz plz help...​

Answer 1

Answer:-

Your Answer Is Option (a) 4.

Explanation:-

Given:-

• A rational number $\bf\dfrac{23457}{2^3\times 5^4}.$

To Find:-

• The decimal expansion wil terminate after how many decimal places.

How To Do??

• Here we will convert the denominator which is in the form $\tt 2^n\times 5^m$ to $\tt 2^m\times5^m.$ Where n<m.

Therefore,

$\\ \tt\implies \dfrac{23457}{ {2}^{3} \times {5}^{4} } \rm \: \: can \: be \: \: written \: \: as \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \tt\implies \frac{23457}{{2}^{3} \times {5}^{4} } \times \frac{ 2 }{ 2 } \: \rm \: \: as \: \: \frac{2}{2} = 1 \: and \: \: 1 \times anyno. = same \: no.$

So by solving,

$\\ \tt\mapsto\frac{23457}{ {2}^{3} \times {5}^{4} } \times \dfrac{2}{2} .$

$\\ \tt = \frac{(23457)2}{2^{3} \times 2 \times 5 {}^{4} }.$

$\\ \tt = \frac{46917}{ {2}^{4} \times {5}^{4} } .$

$\\ \tt = \frac{46914}{(2 \times 5) {}^{4} } .$

$\\ \tt = \frac{46914}{10 {}^{4} } .$

$\\ \tt = \frac{46914}{10000}.$

$\\ \tt = 4. \bf \underline{6914}.$

So we can see that the given rational number's decimal expansion terminate aftwr 4 decimal places.

So The Final Answer Is Option (2).