Let's analyze each expression one by one and determine whether it is equivalent or not equivalent to the given expression [tex]\(7^{1} \cdot 49^{\frac{1}{6}}\)[/tex].
1. Expression: [tex]\(7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}}\)[/tex]
- The given expression is [tex]\(7^{1} \cdot 49^{\frac{1}{6}}\)[/tex].
- First, let's rewrite [tex]\(49\)[/tex] as [tex]\(7^2\)[/tex]. Thus, [tex]\(49^{\frac{1}{6}} = (7^2)^{\frac{1}{6}} = 7^{\frac{2}{6}} = 7^{\frac{1}{3}}\)[/tex].
- So, the original expression becomes [tex]\(7^{1} \cdot 7^{\frac{1}{3}} = 7^{1 + \frac{1}{3}} = 7^{\frac{4}{3}}\)[/tex].
- Now, we need to compare this with [tex]\(7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}}\)[/tex].
- Rewriting [tex]\(49\)[/tex] as [tex]\(7^2\)[/tex], [tex]\(49^{\frac{7}{5}} = (7^2)^{\frac{7}{5}} = 7^{\frac{14}{5}}\)[/tex].
- Thus, [tex]\(7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}} = 7^{\frac{1}{5}} \cdot 7^{\frac{14}{5}} = 7^{\frac{1}{5} + \frac{14}{5}} = 7^{3}\)[/tex].
Since [tex]\(7^{\frac{4}{3}} \ne 7^{3}\)[/tex], the expression is Not Equivalent.
2. Expression: 49
- From the analysis above, the given expression simplifies to [tex]\(7^{\frac{4}{3}}\)[/tex].
- Comparing this with [tex]\(49\)[/tex], and since [tex]\(49 = 7^2\)[/tex], we have [tex]\(7^{\frac{4}{3}} \ne 7^2\)[/tex].
Therefore, the expression is Not Equivalent.
3. Expression: [tex]\(7^{\frac{1}{5}} \cdot 7^{\frac{14}{5}}\)[/tex]
- The original expression [tex]\(7^{1} \cdot 49^{\frac{1}{6}}\)[/tex] simplifies to [tex]\(7^{\frac{4}{3}}\)[/tex] as discussed.
- Now compare with [tex]\(7^{\frac{1}{5}} \cdot 7^{\frac{14}{5}}\)[/tex].
- [tex]\(7^{\frac{1}{5}} \cdot 7^{\frac{14}{5}} = 7^{\frac{1}{5} + \frac{14}{5}} = 7^{\frac{15}{5}} = 7^{3}\)[/tex].
Clearly, [tex]\(7^{\frac{4}{3}} \ne 7^{3}\)[/tex], so the expression is Not Equivalent.
4. Expression: [tex]\(499_{10}^{\frac{2}{10}} \cdot 7^{\frac{1}{5}}\)[/tex]
- Given the original expression simplifies to [tex]\(7^{\frac{4}{3}}\)[/tex] as discussed.
- Comparing this with [tex]\(499_{10}^{\frac{2}{10}} \cdot 7^{\frac{1}{5}}\)[/tex].
- Since [tex]\(499_{10}^{\frac{2}{10}} = 499^{0.2}\)[/tex] is a uniquely different number that doesn’t simplify to a form involving just the same base, it adds complexity that does not align equivalently.
Hence, this expression is also Not Equivalent.
Summary:
\begin{tabular}{|c|c|c|c|}
\hline [tex]$7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}}$[/tex] & Not Equivalent & \\
\hline 49 & Not Equivalent & \\
\hline [tex]$7^{\frac{1}{5}} \cdot 7^{\frac{14}{5}}$[/tex] & Not Equivalent & \\
\hline [tex]$499^{\frac{2}{10}} \cdot 7^{\frac{1}{5}}$[/tex] & Not Equivalent & \\
\hline
\end{tabular}
All given expressions are Not Equivalent to [tex]$7^{1} \cdot 49^{\frac{1}{6}}$[/tex].
