Question

which of the following values is not an irrational number?\\(\boldsymbol{\pi}\\), \\(\boldsymbol{\sqrt3{48}}\\), \\(\boldsymbol{9.8\overline{3}}\\), \\(\boldsymbol{\sqrt{80}}\\)\
options: \\(9.8\overline{3}\\), \\(\pi\\), \\(\sqrt3{48}\\), \\(\sqrt{80}\\)

Explanation:

Step1: Recall irrational number definition

Irrational numbers are non - repeating, non - terminating decimals and cannot be expressed as a fraction of two integers. Rational numbers can be expressed as fractions or are terminating/repeating decimals.

Step2: Analyze \(9.\overline{83}\)

The number \(9.\overline{83}\) is a repeating decimal (the "83" repeats indefinitely). A repeating decimal can be expressed as a fraction. Let \(x = 9.\overline{83}=9.838383\cdots\). Then \(100x=983.838383\cdots\). Subtract \(x\) from \(100x\): \(100x - x=983.8383\cdots - 9.8383\cdots\), \(99x = 974\), so \(x=\frac{974}{99}\), which is a rational number.

Step3: Analyze \(\pi\)

\(\pi = 3.1415926\cdots\) is a non - repeating, non - terminating decimal, so it is irrational.

Step4: Analyze \(\sqrt[3]{48}\)

\(\sqrt[3]{48}=\sqrt[3]{16\times3}=2\sqrt[3]{6}\). Since \(\sqrt[3]{6}\) is non - repeating and non - terminating in its decimal expansion, \(\sqrt[3]{48}\) is irrational.

Step5: Analyze \(\sqrt{80}\)

\(\sqrt{80}=\sqrt{16\times5} = 4\sqrt{5}\). \(\sqrt{5}=2.2360679\cdots\) is non - repeating and non - terminating, so \(\sqrt{80}\) is irrational.

Answer:

\(9.\overline{83}\) (the option with \(9.\overline{83}\))