Question

Choose the option that has at least one irrational number.

• A. $\sqrt[3]{3216},\frac{1}{9}$
• B. $\frac{–13}{9},\sqrt{12\times 27}$
• C. $\frac{3\pi }{16\pi },\frac{29}{7}$
• D. $\sqrt{45},\sqrt{2\times 98}$

Answer 1

The correct option is D $\sqrt{45},\sqrt{2\times 98}$

Irrational numbers have non-terminating and non-repeating decimal expansions.

$\frac{–13}{9},\sqrt{12\times 27}$:

$\frac{–13}{9}=–\mathrm{1.4444...}$ is a non-terminating and repeating decimal and hence a rational number.

$\sqrt{12\times 27}=\sqrt{324}=\sqrt{18\times 18}=18$ is a rational number.

$\frac{3\pi }{16\pi },\frac{29}{7}$:

$\frac{29}{7}=\mathrm{4.142857142...}$ is a non-terminating and repeating decimal and hence a rational number.

$\frac{3\pi }{16\pi }=\frac{3}{16}=0.1875$
is a terminating decimal.

$\sqrt{45},\sqrt{2\times 98}$:

$\sqrt{45}=\sqrt{9\times 5}=3\sqrt{5}$ is an irrational number as $\sqrt{5}$ is an irrational number.

$\sqrt{2\times 98}=\sqrt{196}=14$ is a rational number.

$\sqrt[3]{3216},\frac{1}{9}$:

$\sqrt[3]{3216}=\sqrt[3]{36^3}=6$ is a rational number.

$\frac{1}{9}=\mathrm{0.111...}$ is a non-terminating and repeating decimal and hence a rational number.

Hence, $\sqrt{45},\sqrt{2\times 93}$ is the correct answer.