Question

Fractions in the simplest form that have denominators of 2, 4,8,16, and 32 produce terminating decimals.Fractions with denominators Of 6,12,18, and 24 produce repeating decimals. What causes the difference? Explain.

Answer 1

As you have read that if in a fraction the denominator is of the form $2^{m} \text{ or }  5^{n} \text{ or } 2^{m} \cdot 5^{n}$ that fraction is terminating where m and n are positive integers.

Otherwise if in a fraction the denominator is not of the form  $2^{m} \text{ or } 5^{n}  \text{ or }  2^{m} \cdot 5^{n}$than the fraction is non terminating repeating.

So rational number can be terminating as well as non terminating repeating.

So, you can see as in 2,4 ,8,16,32 the denominator is of the form $2^{1},2^{2},2^{3},2^{4},2^{5}$. That's why the decimal terminates.

Now look at fractions having denominators 6,12,18,24. Now among all these numbers

$6=2 \cdot 3\\12=2\cdot2\cdot3\\18=2\cdot3\cdot3\\24=2\cdot2\cdot2\cdot3$

The prime factor of all these numbers contain factor other than 2 i.e 3 also.

As i have written above if the denominator is not of the form  $2^{m}  \text{ or  } 5^{n} \text{ or  }  2^{m} \cdot 5^{n}$then the decimal is non terminating.

Answer 2

Fractions having a denominator which can be factored in terms of 2 or 5 (or both) will terminate completely or will have terminating decimals.  

But the fractions whose denominator can'not be factored only in terms of 2, 5 or both, will repeat or will have repeating decimals..

As 2= $2^1$ , 4=$2^2$ , 8=$2^3$ , 16=$2^4$

and 32$=2^5$.

Here, 2,4,8,16, and 32 are all powers of 2 and fractions with these denominators will have terminating decimal expansion.  

As,

$6 = 2 \times 3$ ,$12 = 3 \times 4$ ,$18 = 6 \times 3$ ,$24 = 6 \times 4$

But 6,12, 18, and 24 can'not be factored only in terms of 2 or 5 or both. Therefore, fractions with these denominators will have repeating decimals.