Cescalante1286idn
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Select all the correct locations on the table.

Consider the following expression:
[tex]\[ 7^{\frac{1}{1}} \cdot 49^{\frac{1}{6}} \][/tex]

Select "Equivalent" or "Not Equivalent" to indicate whether the expression above is equivalent or not equivalent to the values or expressions in the last column.

[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
\text{Expression} & \text{Equivalent} & \text{Not Equivalent} & \text{Value/Expression} \\
\hline
7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}} & & & 343 \\
\hline
7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}} & & & 49 \\
\hline
7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}} & & & 7^{\frac{1}{5}} \cdot 7^{\frac{14}{5}} \\
\hline
7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}} & & & 499_{10}^{\frac{2}{10}} \cdot 7^{\frac{1}{5}} \\
\hline
\end{tabular}
\][/tex]

Sagot :

GinnyAnsweridn
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].
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