Question

Which of the following numbers represents an irrational number?

• A. $\sqrt[3]{30.343}$
• B. $\sqrt[3]{3216}$
• C. $\sqrt[3]{3\frac{875}{7}}$
• D. $\sqrt[3]{30.0125}$

Answer 1

The correct option is D $\sqrt[3]{30.0125}$

Detailed step-by-step solution:
$\sqrt[3]{30.343}=\sqrt[3]{3\frac{343}{1000}}$
$=\frac{\sqrt[3]{3343}}{\sqrt[3]{31000}}$
$=\frac{\sqrt[3]{37\times 7\times 7}}{\sqrt[3]{310\times 10\times 10}}$
$=\frac{7}{10}=0.7$
0.7 is a rational number.
$\sqrt[3]{3\frac{875}{7}}=\sqrt[3]{3125}$
$=\sqrt[3]{35\times 5\times 5}$
= 5
5 is a rational number.
$\sqrt[3]{30.0125}=\sqrt[3]{3\frac{125}{10000}}$
$=\frac{\sqrt[3]{3125}}{\sqrt[3]{310000}}$
10,000 is not a perfect cube. So, $\sqrt[3]{310000}$ is an irrational number.
So, $\frac{5}{\sqrt[3]{310000}}$ or $\sqrt[3]{30.0125}$ is also an irrational number.
$\sqrt[3]{3216}=\sqrt[3]{36\times 6\times 6}=6$
6 is a rational number.
So, option C is the correct answer.