Let's analyze the given expression step-by-step to determine its equivalence to the provided values and expressions.
We're given:
[tex]\[ 7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}} \][/tex]
First, let's simplify [tex]\( 49^{\frac{7}{5}} \)[/tex]:
[tex]\[ 49 = 7^2 \][/tex]
So,
[tex]\[ 49^{\frac{7}{5}} = (7^2)^{\frac{7}{5}} = 7^{\frac{14}{5}} \][/tex]
Now, combining both terms using the properties of exponents:
[tex]\[ 7^{\frac{1}{5}} \cdot 7^{\frac{14}{5}} \][/tex]
We apply the property of exponents [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex]:
[tex]\[ 7^{\frac{1}{5} + \frac{14}{5}} = 7^{\frac{15}{5}} = 7^3 \][/tex]
Simplifying further:
[tex]\[ 7^3 = 343 \][/tex]
Let's examine each expression one by one for equivalence:
1. [tex]\( 343 \)[/tex]
[tex]\[ 7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}} = 343 \][/tex]
Equivalent: Yes, it is equivalent to [tex]\( 343 \)[/tex].
2. [tex]\( 49 \)[/tex]
[tex]\[ 7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}} = 49 \][/tex]
Not Equivalent: No, it is not equivalent to [tex]\( 49 \)[/tex].
3. [tex]\( 7^{\frac{1}{5}} \cdot 7^{\frac{14}{5}} \)[/tex]
[tex]\[ 7^{\frac{1}{5}} \cdot 7^{\frac{14}{5}} = 7^3 \][/tex]
Not Equivalent: This looks equivalent in form but directly [tex]\[ 7^{\frac{1}{5}} \cdot 7^{\frac{14}{5}} \][/tex] needs to be simplified to see it equals [tex]\( 7^3 \)[/tex] which is [tex]\(343\)[/tex].
4. [tex]\( 49^{\frac{2}{10}} \cdot 7^{\frac{1}{5}} \)[/tex]
Since [tex]\( 49 = 7^2 \)[/tex], this can be rewritten:
[tex]\[ 49^{\frac{2}{10}} = (7^2)^{\frac{1}{5}} = 7^{\frac{2}{5}} \][/tex]
Our expression becomes:
[tex]\[ 7^{\frac{2}{5}} \cdot 7^{\frac{1}{5}} \][/tex]
Which simplifies to:
[tex]\[ 7^{\frac{3}{5}} \neq 7^3 \][/tex]
Not Equivalent: No, it is not equivalent because 7^{\frac{2}{5}} \] does not simplify to [tex]\(343\)[/tex].
Now we compile our results into the provided table format:
\begin{tabular}{|c|c|c|c|}
\hline
Expression & Equivalent & Not Equivalent & Value/Expression \\
\hline
[tex]\( 7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}} \)[/tex] & Equivalent & & [tex]\( 343 \)[/tex] \\
\hline
[tex]\( 7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}} \)[/tex] & & Not Equivalent & [tex]\( 49 \)[/tex] \\
\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}
This explains all parts of the given problem in detail.
