Let's carefully analyze each expression to determine whether it is equivalent to the given expression [tex]\(7^{\frac{1}{8}} \cdot 49^{\frac{7}{6}}\)[/tex].
### Original Expression
The expression we are comparing to is:
[tex]\[ 7^{\frac{1}{8}} \cdot 49^{\frac{7}{6}} \][/tex]
### Comparison with Other Expressions
1. Expression [tex]\[7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}}\][/tex]
- This needs to be compared with [tex]\(7^{\frac{1}{8}} \cdot 49^{\frac{7}{6}}\)[/tex].
- The bases and exponents are different; hence, it is not equivalent.
2. Expression [tex]\[7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}}\][/tex]
- This is exactly the same expression as in the first row, so we can conclude it's not equivalent.
3. Expression [tex]\[7^{\frac{1}{5}} \cdot 7^{\frac{14}{5}}\][/tex]
- Simplify the expression: [tex]\(7^{\frac{1}{5}} \cdot 7^{\frac{14}{5}} = 7^{\left(\frac{1}{5} + \frac{14}{5}\right)} = 7^{3}\)[/tex].
- Since [tex]\(49 = 7^2\)[/tex], we can see that [tex]\(49^{7/6} \neq 7^3\)[/tex] in the context of the original expression, and thus they are not equivalent.
4. Expression [tex]\[49^{\frac{2}{10}} \cdot 7^{\frac{1}{5}}\][/tex]
- Simplify the exponent [tex]\( \frac{2}{10} = \frac{1}{5} \)[/tex], so the expression becomes [tex]\(49^{\frac{1}{5}} \cdot 7^{\frac{1}{5}}\)[/tex].
- Again comparing with our original expression, we see that these are not equivalent.
### Final Table with Judgements on Equivalency:
\begin{tabular}{|c|c|c|c|}
\hline
[tex]$7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}}$[/tex] & \ & ✔ Not Equivalent & 343 \\
\hline
[tex]$7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}}$[/tex] & & ✔ Not Equivalent & 49 \\
\hline
[tex]$7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}}$[/tex] & & ✔ Not Equivalent & [tex]$7^{\frac{1}{5}} \cdot 7^{\frac{14}{5}}$[/tex] \\
\hline
[tex]$7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}}$[/tex] & & ✔ Not Equivalent & [tex]$49^{\frac{2}{10}} \cdot 7^{\frac{1}{5}}$[/tex] \\
\hline
\end{tabular}
In conclusion, all the given expressions are "Not Equivalent" to [tex]\(7^{\frac{1}{8}} \cdot 49^{\frac{7}{6}}\)[/tex].
