Let's solve each part of the question step-by-step.
### Part (a) - Difference of [tex]\(2 \sqrt{2} - \sqrt{2}\)[/tex]
First, let's compute the difference:
[tex]\[ 2 \sqrt{2} - \sqrt{2} \][/tex]
We can factor out the common term, [tex]\(\sqrt{2}\)[/tex]:
[tex]\[ 2 \sqrt{2} - \sqrt{2} = (2 - 1) \sqrt{2} \][/tex]
[tex]\[ = 1 \cdot \sqrt{2} \][/tex]
[tex]\[ = \sqrt{2} \][/tex]
Since [tex]\(\sqrt{2}\)[/tex] is an irrational number (as it cannot be expressed as a fraction of two integers or a terminating decimal), this means that the difference [tex]\(2 \sqrt{2} - \sqrt{2}\)[/tex] is also irrational. Hence,
[tex]\[
\boxed{\text{irrational}}
\][/tex]
### Part (b) - Product of [tex]\(\sqrt{2} \times 2 \sqrt{2}\)[/tex]
Now, let's compute the product:
[tex]\[ \sqrt{2} \times 2 \sqrt{2} \][/tex]
Firstly, rewrite it as:
[tex]\[ \sqrt{2} \times 2 \times \sqrt{2} \][/tex]
Combine the [tex]\(\sqrt{2}\)[/tex] terms:
[tex]\[ = 2 \times (\sqrt{2} \times \sqrt{2}) \][/tex]
We know that [tex]\(\sqrt{2} \times \sqrt{2} = 2\)[/tex]. Therefore:
[tex]\[ 2 \times 2 = 4 \][/tex]
Since 4 is a rational number (because it can be expressed as [tex]\(\frac{4}{1}\)[/tex]), the product [tex]\(\sqrt{2} \times 2 \sqrt{2}\)[/tex] is rational. Hence,
[tex]\[
\boxed{\text{rational}}
\][/tex]
### Part (c) - Quotient of [tex]\(\sqrt{2} \div 2 \sqrt{2}\)[/tex]
Next, let's compute the quotient:
[tex]\[ \frac{\sqrt{2}}{2 \sqrt{2}} \][/tex]
Simplify the fraction:
[tex]\[ = \frac{\sqrt{2}}{2} \times \frac{1}{\sqrt{2}} \][/tex]
[tex]\[ = \frac{\sqrt{2} \times 1}{2 \times \sqrt{2}} \][/tex]
[tex]\[ = \frac{\sqrt{2}}{2 \sqrt{2}} \][/tex]
[tex]\[ = \frac{1}{2} \][/tex]
[tex]\[ = 0.5 \][/tex]
Since 0.5 is a rational number (because it can be expressed as [tex]\(\frac{1}{2}\)[/tex]), the quotient [tex]\(\frac{\sqrt{2}}{2 \sqrt{2}}\)[/tex] is rational. Hence,
[tex]\[
\boxed{\text{rational}}
\][/tex]
