Question

miko multiplied \\(\frac{4}{3}\\) by another number. the product was an irrational number.
select all the numbers that could have been mikos other number.
\\(\square \frac{3}{4}\\) \\(\square \frac{\pi}{4}\\) \\(\square \frac{5}{4}\\) \\(\square \sqrt{3}\\) \\(\square \pi\\) \\(\square \sqrt{4}\\)

Explanation:

Step1: Recall Rational/Irrational Properties

A rational number is a fraction of integers (denominator ≠ 0), or a terminating/repeating decimal. An irrational number is non - repeating, non - terminating, like \( \pi \), \( \sqrt{3} \) (since 3 is not a perfect square). When multiplying a rational number (\( \frac{4}{3} \) is rational) by another number:

• If we multiply by a rational number, the product is rational (because \( \frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd} \), where \( a,b,c,d \) are integers, \( b,d

eq0 \)).

• If we multiply by an irrational number, the product is irrational (except when the rational number is 0, but \( \frac{4}{3}

eq0 \)).

Step2: Analyze Each Option

• For \( \frac{3}{4} \): It is rational. \( \frac{4}{3}\times\frac{3}{4} = 1 \), which is rational. So this is not a valid number.
• For \( \frac{\pi}{4} \): \( \pi \) is irrational, so \( \frac{\pi}{4} \) is irrational. \( \frac{4}{3}\times\frac{\pi}{4}=\frac{\pi}{3} \), which is irrational. So this is a valid number.
• For \( \frac{5}{4} \): It is rational. \( \frac{4}{3}\times\frac{5}{4}=\frac{5}{3} \), which is rational. So this is not a valid number.
• For \( \sqrt{3} \): \( \sqrt{3} \) is irrational. \( \frac{4}{3}\times\sqrt{3}=\frac{4\sqrt{3}}{3} \), which is irrational. So this is a valid number.
• For \( \pi \): \( \pi \) is irrational. \( \frac{4}{3}\times\pi=\frac{4\pi}{3} \), which is irrational. So this is a valid number.
• For \( \sqrt{4} \): \( \sqrt{4} = 2 \), which is rational. \( \frac{4}{3}\times2=\frac{8}{3} \), which is rational. So this is not a valid number.

Answer:

\( \frac{\pi}{4} \), \( \sqrt{3} \), \( \pi \) (the checkboxes corresponding to \( \frac{\pi}{4} \), \( \sqrt{3} \), and \( \pi \) should be selected)