Question

which fractions represent irrational numbers? select two answer. \\(\frac{\sqrt{3}}{16}\\) \\(\frac{16}{3}\\) \\(\frac{16\pi}{3}\\) \\(\frac{\sqrt{16}}{3}\\) \\(\frac{3}{16}\\)

Explanation:

Step1: Recall irrational number definition

Irrational numbers are non - repeating, non - terminating decimals and cannot be expressed as a ratio of two integers ($\frac{p}{q}, q
eq0$) in simplest form if the numerator involves an irrational number.

Step2: Analyze each option

• Option 1: $\frac{\sqrt{3}}{16}$: $\sqrt{3}$ is an irrational number (since 3 is not a perfect square). When we divide an irrational number by a non - zero rational number (16 is rational), the result is still irrational.
• Option 2: $\frac{16}{3}$: 16 and 3 are both integers, and this is a rational number (it can be expressed as a ratio of two integers).
• Option 3: $\frac{16\pi}{3}$: $\pi$ is an irrational number. When we multiply an irrational number ($\pi$) by a non - zero rational number (16 and 3 are rational, so $\frac{16}{3}$ is rational) and then divide (or multiply, in this case multiply $\pi$ by $\frac{16}{3}$), the result is irrational.
• Option 4: $\frac{\sqrt{16}}{3}$: $\sqrt{16} = 4$, which is an integer. So $\frac{\sqrt{16}}{3}=\frac{4}{3}$, and this is a rational number (ratio of two integers).
• Option 5: $\frac{3}{16}$: 3 and 16 are integers, so this is a rational number (ratio of two integers).

Answer:

$\frac{\sqrt{3}}{16}$, $\frac{16\pi}{3}$