Step 1

Given the fraction

a) \(\displaystyle{\frac{{{72}}}{{{81}}}}\)

A rational number \(\displaystyle{\frac{{{a}}}{{{b}}}}\) in the simplest form can be written as a terminating decimal if an only if the prime factorization of the denominator contains no primes other than 2 or 5.

Step 2

Here first simplify the given fraction

\(\displaystyle={\frac{{{72}}}{{{81}}}}\)

\(\displaystyle={\frac{{{2}\times{2}\times{2}\times{3}\times{3}}}{{{3}\times{3}\times{3}\times{3}}}}\).

\(\displaystyle={\frac{{{8}}}{{{3}^{{{2}}}}}}\)

The denominator is 3

Step 3

Therefore,

No the denominator of the simplified fraction contains a factor other than 2 or 5

Step 4

b) \(\displaystyle{\frac{{{21}}}{{{28}}}}\)

Here first simplify the given fraction

\(\displaystyle={\frac{{{3}\times{7}}}{{{2}\times{2}\times{7}}}}\)

\(\displaystyle={\frac{{{3}}}{{{2}^{{{3}}}}}}\)

Here the denominator is 2

Step 5

yes the only factors of the denominator of the simplified fraction is 2

Given the fraction

a) \(\displaystyle{\frac{{{72}}}{{{81}}}}\)

A rational number \(\displaystyle{\frac{{{a}}}{{{b}}}}\) in the simplest form can be written as a terminating decimal if an only if the prime factorization of the denominator contains no primes other than 2 or 5.

Step 2

Here first simplify the given fraction

\(\displaystyle={\frac{{{72}}}{{{81}}}}\)

\(\displaystyle={\frac{{{2}\times{2}\times{2}\times{3}\times{3}}}{{{3}\times{3}\times{3}\times{3}}}}\).

\(\displaystyle={\frac{{{8}}}{{{3}^{{{2}}}}}}\)

The denominator is 3

Step 3

Therefore,

No the denominator of the simplified fraction contains a factor other than 2 or 5

Step 4

b) \(\displaystyle{\frac{{{21}}}{{{28}}}}\)

Here first simplify the given fraction

\(\displaystyle={\frac{{{3}\times{7}}}{{{2}\times{2}\times{7}}}}\)

\(\displaystyle={\frac{{{3}}}{{{2}^{{{3}}}}}}\)

Here the denominator is 2

Step 5

yes the only factors of the denominator of the simplified fraction is 2