Question

classify each number below as a rational number or an irrational number.

	rational	irrational
-√16	○	○
-71.\overline{28}	○	○
√2	○	○
7π	○	○
-66.18	○	○

Explanation:

Step1: Analyze \(-\sqrt{16}\)

Simplify \(-\sqrt{16}\): \(\sqrt{16} = 4\), so \(-\sqrt{16}=- 4\). Integers are rational numbers (as they can be written as \(\frac{-4}{1}\)). So \(-\sqrt{16}\) is rational.

Step2: Analyze \(-71.\overline{28}\)

Repeating decimals are rational because they can be expressed as a fraction. Let \(x=-71.\overline{28}\), then \(100x=-7128.\overline{28}\), subtracting \(x\) from \(100x\) gives \(99x=-7128 + 71=-7057\), so \(x =-\frac{7057}{99}\). Thus, \(-71.\overline{28}\) is rational.

Step3: Analyze \(\sqrt{2}\)

\(\sqrt{2}\) is a non - repeating, non - terminating decimal (its decimal expansion is \(1.41421356\cdots\)) and cannot be expressed as a fraction of two integers. So \(\sqrt{2}\) is irrational.

Step4: Analyze \(7\pi\)

\(\pi\) is irrational (\(\pi = 3.14159265\cdots\)), and multiplying an irrational number by a non - zero integer (7) gives an irrational number. So \(7\pi\) is irrational.

Step5: Analyze \(-66.18\)

Terminating decimals are rational because they can be written as a fraction (e.g., \(-66.18=-\frac{6618}{100}=-\frac{3309}{50}\)). So \(-66.18\) is rational.

Answer:

• For \(-\sqrt{16}\): Mark the "rational" circle.
• For \(-71.\overline{28}\): Mark the "rational" circle.
• For \(\sqrt{2}\): Mark the "irrational" circle.
• For \(7\pi\): Mark the "irrational" circle.
• For \(-66.18\): Mark the "rational" circle.