Questions: Select all the correct locations on the table. Consider the following expression. [ 7^1/5 cdot 49^1/t ] 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. - 7^1/5 cdot 49^7/5 Equivalent Not Equivalent 343 - 7^1/5 cdot 49^7/5 Equivalent Not Equivalent 49 - 7^1/5 cdot 49^7/5 Equivalent Not Equivalent 7^1/5 cdot 7^14/5 - 7 frac15 cdot 49^7/5 Equivalent Not Equivalent 49 frac210 cdot 7^1/5

$Select all the correct locations on the table. Consider the following expression. [ 7^1/5 cdot 49^1/t ] Select "equivalent" or "not equivalent" to indicate whether the expression above is equivalent or not equivalent to the values or expressions in the last column. - 7^1/5 cdot 49^7/5 Equivalent Not Equivalent 343 - 7^1/5 cdot 49^7/5 Equivalent Not Equivalent 49 - 7^1/5 cdot 49^7/5 Equivalent Not Equivalent 7^1/5 cdot 7^14/5 - 7 frac15 cdot 49^7/5 Equivalent Not Equivalent 49 frac210 cdot 7^1/5$

Transcript text: Select all the correct locations on the table. Consider the following expression. \[ 7^{\frac{1}{5}} \cdot 49^{\frac{1}{t}} \] 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. \begin{tabular}{|c|c|c|c|} \hline $7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}}$ & Equivalent & Not Equivalent & 343 \\ \hline $7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}}$ & Equivalent & Not Equivalent & 49 \\ \hline $7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}}$ & Equivalent & Not Equivalent & $7^{\frac{1}{5}} \cdot 7^{\frac{14}{5}}$ \\ \hline $7 \frac{1}{5} \cdot 49^{\frac{7}{5}}$ & Equivalent & Not Equivalent & $49 \frac{2}{10} \cdot 7^{\frac{1}{5}}$ \\ \hline \end{tabular}

Solution

Solution Steps

Step 1: Simplifying the Original Expression

We start with the expression $7^{\frac{1}{5}} \cdot 49^{\frac{1}{t}}$. Recognizing that $49$ can be expressed as $7^2$, we rewrite the expression as: \[ 7^{\frac{1}{5}} \cdot (7^2)^{\frac{1}{t}} = 7^{\frac{1}{5}} \cdot 7^{\frac{2}{t}} = 7^{\frac{1}{5} + \frac{2}{t}} \]

Step 2: Evaluating Each Expression in the Table

Next, we evaluate the expressions in the table to see if they are equivalent to the original expression.

For $343$: \[ 343 = 7^3 \quad \text{(since $7^3 = 343$)} \] We need to check if $\frac{1}{5} + \frac{2}{t} = 3$.
For $49$: \[ 49 = 7^2 \quad \text{(since $7^2 = 49$)} \] We need to check if $\frac{1}{5} + \frac{2}{t} = 2$.
For $7^{\frac{1}{5}} \cdot 7^{\frac{14}{5}}$: \[ 7^{\frac{1}{5}} \cdot 7^{\frac{14}{5}} = 7^{\frac{1}{5} + \frac{14}{5}} = 7^{3} \] We need to check if $\frac{1}{5} + \frac{2}{t} = 3$.

Step 3: Checking Equivalence

From the evaluations:

For $343$, we found that it is equivalent since both simplify to $7^3$.
For $49$, it is also equivalent since both simplify to $7^2$.
For $7^{\frac{1}{5}} \cdot 7^{\frac{14}{5}}$, it is equivalent as it simplifies to $7^3$.