Which of the following fractions will terminate when expressed as a decimal? (Choose all that apply.)
Recall that in order for the decimal expansion of a fraction to terminate, the fraction's denominator, in fully reduced form, must have a prime factorization that consists of only $$2$$'s and/or $$5$$'s.
The denominator in $$A$$ is composed of only $$2$$'s, $$(256 = 2^8)$$.
The denominator in $$B$$ is composed of only $$2$$'s and $$5$$'s, $$(100=2^2\times 5^2)$$.
In fully reduced form, the fraction in $$D$$ is equal to $$\dfrac{7}{20}$$ and the denominator $$20$$ is composed of only $$2$$'s and $$5$$'s $$(20=2^2\times 5)$$.
By contrast, the denominator in $$C$$ has prime factors other than $$2$$'s and $$5$$'s $$(27 =3^3)$$, and in fully reduced form, the fraction in $$E$$ is equal to $$\dfrac{1}{15}$$, and $$15$$ has a prime factor other than $$2$$'s and $$5$$'s $$(15 = 3 \times 5)$$.
Therefore, options $$A$$, $$B$$ and $$D$$ are the correct answers.
Look at several examples of rational numbers in the form $$\displaystyle\frac{p}{q}(q\neq 0)$$, where $$p$$ and $$q$$ are integers with no common factors other than $$1$$ and having terminating decimal representaions (expansions). Can you guess what property $$q$$ must satisfy?
If $$\displaystyle d=\frac { 1 }{ { 2 }^{ 3 }\times { 5 }^{ 7 } } $$ is expressed as a terminating decimal, how many non zero digits will $$d$$ have?
Without actually performing the long division, state whether the following rational number have terminating or non-terminating repeating (recurring) decimal expansion: $$\frac{3}{8}$$
Without actually performing the long division, state whether the following rational number has terminating or non-terminating repeating (recurring) decimal expansion: $$\dfrac{29}{343}$$
Without actually performing the long division, state whether the following rational number have terminating or non-terminating repeating (recurring) decimal expansion: $$\dfrac{13}{125}$$
Without actually performing the long division, state whether the following rational number have terminating or non-terminating repeating (recurring) decimal expansion: $$\dfrac{7}{80}$$
Without actually performing the long division, state whether the following rational number have terminating or non-terminating repeating (recurring) decimal expansion: $$\frac{64}{455}$$
Without actually performing the long division, state whether the following rational number have terminating or non-terminating repeating (recurring) decimal expansion: $$\frac{6}{15}$$
Without actually performing the long division, state whether the following rational number have terminating or non-terminating repeating (recurring) decimal expansion: $$\frac{35}{50}$$
Without actually performing the long division, state whether the following rational number has terminating or non-terminating repeating (recurring) decimal expansion: $$\dfrac{129}{2^{2}*5^{7}*7^{5}}$$