Sure, let's solve the problem step by step.
We need to determine whether the number \( \frac{\sqrt{2}}{3 \sqrt{8}} \) is rational or irrational.
1. Simplify the Denominator:
We start with the denominator: \( 3 \sqrt{8} \).
Notice that \( \sqrt{8} \) can be simplified as follows:
[tex]\[
\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2 \sqrt{2}
\][/tex]
Substituting this back into our expression, the denominator becomes:
[tex]\[
3 \sqrt{8} = 3 \times 2 \sqrt{2} = 6 \sqrt{2}
\][/tex]
2. Form the Fraction:
Now, we replace \( \sqrt{8} \) in the denominator of our original fraction:
[tex]\[
\frac{\sqrt{2}}{3 \sqrt{8}} = \frac{\sqrt{2}}{6 \sqrt{2}}
\][/tex]
3. Simplify the Fraction:
Next, we notice that both the numerator and the denominator have a factor of \( \sqrt{2} \). We can cancel this common factor:
[tex]\[
\frac{\sqrt{2}}{6 \sqrt{2}} = \frac{1}{6}
\][/tex]
4. Determine the Nature of the Result:
The simplified fraction is \( \frac{1}{6} \), which is a rational number because it can be expressed as the ratio of two integers, 1 and 6, with 6 not being zero.
Therefore, \( \frac{\sqrt{2}}{3 \sqrt{8}} \) simplifies to \( \frac{1}{6} \), which is a rational number.
So, the number [tex]\( \frac{\sqrt{2}}{3 \sqrt{8}} \)[/tex] is [tex]\(\textbf{rational}\)[/tex].
