Question

without actually performing the long division state whether 13/3125 and 13/343 will have a terminating decimal expansion or a non - terminating repeating decimal decimal expansion?​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

$\displaystyle \sf{\: Fraction \: = \: \frac{Numerator}{Denominator} \: }$

A fraction is said to have terminating decimal expansion if the 2 & 5 are the only prime factors of the denominator

Otherwise the fraction is said to have non terminating decimal expansion

TO CHECK

To identify the below two fractions have terminating decimal expansion or a non - terminating repeating decimal decimal expansion

$\displaystyle \sf{ (i) \: \: \: \frac{13}{3125} \: \: }$

$\displaystyle \sf{ (ii) \: \: \: \frac{13}{343} \: \: }$

CALCULATION

CHECKING FOR OPTION (i)

$\displaystyle \sf{ (i) \: \: \: \frac{13}{3125} \: \: }$

Here denominator = 3125

Now

$\sf{ 3125 = 5 \times 5 \times 5 \times 5 \times 5\: \: }$

Since 5 is the only prime number present in prime factorisation of 3125

Hence this fraction has terminating decimal expansion

CHECKING FOR OPTION (ii)

$\displaystyle \sf{ (ii) \: \: \: \frac{13}{343} \: \: }$

Here denominator = 343

Now

$\sf{ 343 = 7 \times 7 \times 7\: }$

Since 7 is the prime number present in prime factorisation of 343

Hence this fraction has non terminating decimal expansion

RESULT

$\boxed{ \displaystyle \sf{ (i) \: \frac{13}{3125} \: \: has \: terminating \: decimal \: expansion \: }}$

$\boxed{ \displaystyle \sf{ (ii) \: \frac{13}{343} \: \: has \:non \: terminating \: decimal \: expansion \: }}$

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LEARN MORE FROM BRAINLY

Out of the following which are proper fractional numbers

(i)3/2

(ii)2/5

(iii)1/7

(iv)8/3

https://brainly.in/question/4865271

Answer 2

Answer:

Shown the Example of One Question Now you try the second one Hope you can understand the concept ..Best of luck