Question

for each number below, choose the best description of its decimal expansion.
(a) $\frac{1}{3}$
\bigcirc terminating
\bigcirc repeating
\bigcirc neither terminating nor repeating
\bigcirc does not exist

(b) $2\pi$
\bigcirc terminating
\bigcirc repeating
\bigcirc neither terminating nor repeating
\bigcirc does not exist

(c) $-\sqrt{3}$
\bigcirc terminating
\bigcirc repeating
\bigcirc neither terminating nor repeating
\bigcirc does not exist

(d) $\sqrt{0.49}$
\bigcirc terminating
\bigcirc repeating
\bigcirc neither terminating nor repeating

Explanation:

Step1: Analyze $\frac{1}{3}$

When we divide 1 by 3, we get $0.\overline{3}$, a repeating decimal.

Step2: Analyze $2\pi$

$\pi$ is an irrational number with a non-terminating, non-repeating decimal expansion. Multiplying by 2 preserves this property.

Step3: Analyze $-\sqrt{3}$

$\sqrt{3}$ is irrational, so its negative also has a non-terminating, non-repeating decimal expansion.

Step4: Analyze $\sqrt{0.49}$

Calculate the square root: $\sqrt{0.49} = 0.7$, a terminating decimal.

Answer:

(a) Repeating
(b) Neither terminating nor repeating
(c) Neither terminating nor repeating
(d) Terminating