Question

17 is the product of two irrational numbers always irrational? justify your answer

1. is the product of \\(\sqrt{16}\\) and \\(\frac{4}{7}\\) rational or irrational? explain your reasoning.
2. classify the expression \\(\frac{2}{\sqrt{144}} + \frac{\sqrt{169}}{3}\\) as rational or irrational. explain your reasoning.

Explanation:

Question 17 (Assuming the question is "Is the product of two irrational numbers always irrational? Justify your answer")

Step1: Recall the definition of irrational numbers

Irrational numbers are numbers that cannot be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q
eq0$.

Step2: Provide a counter - example

Consider the two irrational numbers $\sqrt{2}$ and $\sqrt{2}$. Their product is $\sqrt{2}\times\sqrt{2} = 2$. Now, 2 can be written as $\frac{2}{1}$, where $p = 2$ and $q = 1$ are integers and $q
eq0$. So, 2 is a rational number. This shows that the product of two irrational numbers is not always irrational.

Step1: Simplify $\sqrt{16}$

We know that $\sqrt{16}=4$ because $4\times4 = 16$.

Step2: Find the product of 4 and $\frac{4}{7}$

The product of 4 (which is $\frac{4}{1}$) and $\frac{4}{7}$ is $\frac{4}{1}\times\frac{4}{7}=\frac{16}{7}$.

Step3: Determine if $\frac{16}{7}$ is rational

A rational number is a number that can be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q
eq0$. Here, $p = 16$ and $q = 7$ are integers and $q
eq0$, so $\frac{16}{7}$ is a rational number.

Step1: Simplify $\sqrt{144}$ and $\sqrt{169}$

We know that $\sqrt{144}=12$ (since $12\times12 = 144$) and $\sqrt{169}=13$ (since $13\times13 = 169$).

Step2: Substitute the simplified values into the expression

The expression $\frac{2}{\sqrt{144}}+\frac{\sqrt{169}}{3}$ becomes $\frac{2}{12}+\frac{13}{3}$.

Step3: Simplify each fraction

Simplify $\frac{2}{12}$ to $\frac{1}{6}$. Now the expression is $\frac{1}{6}+\frac{13}{3}$.

Step4: Find a common denominator and add the fractions

The common denominator of 6 and 3 is 6. So, $\frac{13}{3}=\frac{13\times2}{3\times2}=\frac{26}{6}$. Then $\frac{1}{6}+\frac{26}{6}=\frac{1 + 26}{6}=\frac{27}{6}=\frac{9}{2}$.

Step5: Determine if $\frac{9}{2}$ is rational

Since $\frac{9}{2}$ can be written as a fraction of two integers (9 and 2) with a non - zero denominator, it is a rational number.

Answer:

The product of two irrational numbers is not always irrational. For example, $\sqrt{2}\times\sqrt{2}=2$, and 2 is a rational number.

Question 18