Question

Without dividing, determine which of the following represent terminati

Factors and multiples
ANSWERED
asked 2021-08-17
Without dividing, determine which of the following represent terminating decimals.
1) \(\displaystyle{\frac{{{72}}}{{{81}}}}\)
2) \(\displaystyle{\frac{{{21}}}{{{28}}}}\)
a. Is \(\displaystyle{\frac{{{72}}}{{{81}}}}\) a terminating decimal?
b. Is \(\displaystyle{\frac{{{21}}}{{{28}}}}\)
A) Yes, because the only factors of the denominator, 81, are 2 and 5.
B) No, because the only factors of the denominator, 81, are 2 and 5.
C) No, the denominator of the simplified fraction contains a factor other than 2 or 5.
D) Yes, the only factors of the denominator of the simplified fraction are 2 or 5.

Answers (1)

2021-08-18
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
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