Question

part d
question
select all the expressions that result in a product that is a rational number.
\\(\frac{4}{3} cdot \frac{12}{3}\\) \\(\frac{32}{4} cdot \frac{15}{4}\\)
\\(\frac{sqrt{3}}{2} cdot \frac{22}{7}\\) \\(sqrt{11} cdot \frac{2}{3}\\)

Explanation:

Step1: Recall Rational Number Definition

A rational number is a number that can be expressed as $\frac{p}{q}$, where $p$ and $q$ are integers and $q
eq0$. For a product to be rational, the result should be in this form.

Step2: Analyze $\boldsymbol{\frac{4}{3}\cdot\frac{12}{3}}$

Multiply the numerators and denominators: $\frac{4\times12}{3\times3}=\frac{48}{9}=\frac{16}{3}$. Since $16$ and $3$ are integers ($3
eq0$), this is rational.

Step3: Analyze $\boldsymbol{\frac{32}{4}\cdot\frac{15}{4}}$

Simplify and multiply: $\frac{32\times15}{4\times4}=\frac{480}{16} = 30$. $30$ is an integer (hence rational, as $30=\frac{30}{1}$).

Step4: Analyze $\boldsymbol{\frac{\sqrt{3}}{2}\cdot\frac{22}{7}}$

Multiply: $\frac{\sqrt{3}\times22}{2\times7}=\frac{11\sqrt{3}}{7}$. Since $\sqrt{3}$ is irrational, the product is irrational.

Step5: Analyze $\boldsymbol{\sqrt{11}\cdot\frac{2}{3}}$

Multiply: $\frac{2\sqrt{11}}{3}$. $\sqrt{11}$ is irrational, so the product is irrational.

Answer:

$\frac{4}{3}\cdot\frac{12}{3}$, $\frac{32}{4}\cdot\frac{15}{4}$