Question

which of the following shows that the quotient of two irrational numbers can be irrational? a. \\(sqrt{2} \div 2sqrt{5}\\) b. \\(\frac{pi}{2} \div pi\\) c. \\(2sqrt{5} \div sqrt{5}\\) d. \\(5pi \div pi\\)

Explanation:

Step1: Analyze Option A

Simplify \(\sqrt{2} \div 2\sqrt{5}\). Using the rule of dividing square roots, we can rewrite it as \(\frac{\sqrt{2}}{2\sqrt{5}}\). Rationalizing the denominator (multiplying numerator and denominator by \(\sqrt{5}\)) gives \(\frac{\sqrt{2}\times\sqrt{5}}{2\times5}=\frac{\sqrt{10}}{10}\). \(\sqrt{10}\) is irrational, so \(\frac{\sqrt{10}}{10}\) is also irrational. Both \(\sqrt{2}\) and \(2\sqrt{5}\) are irrational, and their quotient is irrational.

Step2: Analyze Option B

Simplify \(\frac{\pi}{2} \div \pi\). This is \(\frac{\pi}{2}\times\frac{1}{\pi}=\frac{1}{2}\), which is rational.

Step3: Analyze Option C

Simplify \(2\sqrt{5} \div \sqrt{5}\). The \(\sqrt{5}\) terms cancel out, giving \(2\), which is rational.

Step4: Analyze Option D

Simplify \(5\pi \div \pi\). The \(\pi\) terms cancel out, giving \(5\), which is rational.

Answer:

A. \(\boldsymbol{\sqrt{2} \div 2\sqrt{5}}\)