Sure, here is a detailed, step-by-step solution for each missing value in the table.
### Row 1: Calculating the Exponent
Given:
[tex]\[ 70^\text{Exponent} = 7 \][/tex]
To find the exponent:
[tex]\[ \text{Exponent} = \log_{70}(7) \][/tex]
Using logarithm properties, we can express it with natural logarithms:
[tex]\[ \text{Exponent} = \frac{\log(7)}{\log(70)} \][/tex]
From the result:
[tex]\[ \text{Exponent} \approx 0.458 \][/tex]
### Row 2: Calculating the Base
Given:
[tex]\[ \text{Base}^4 = 87 \][/tex]
To find the base:
[tex]\[ \text{Base} = \sqrt[4]{87} \][/tex]
From the result:
[tex]\[ \text{Base} \approx 3.054 \][/tex]
### Row 3: Calculating the Result (Polencia)
Given:
[tex]\[ 4^2 = \text{Result (Polencia)} \][/tex]
To find the result:
[tex]\[ \text{Result (Polencia)} = 4^2 \][/tex]
From the result:
[tex]\[ \text{Result (Polencia)} = 16 \][/tex]
### Row 4: Calculating the Exponent
Given:
[tex]\[ (-3)^\text{Exponent} = 9 \][/tex]
To find the exponent:
We recognize that:
[tex]\[ 9 = (-3)^2 \][/tex]
Therefore:
[tex]\[ \text{Exponent} = 2 \][/tex]
### Row 5: Calculating the Exponent
Given:
[tex]\[ (-2)^\text{Exponent} = -8 \][/tex]
To find the exponent:
We recognize that:
[tex]\[ -8 = (-2)^3 \][/tex]
Therefore:
[tex]\[ \text{Exponent} = 3 \][/tex]
### Row 6: Calculating the Result (Polencia)
Given:
[tex]\[ \left(\frac{2}{3}\right)^4 = \text{Result (Polencia)} \][/tex]
To find the result:
[tex]\[ \text{Result (Polencia)} = \left(\frac{2}{3}\right)^4 \][/tex]
From the result:
[tex]\[ \text{Result (Polencia)} \approx 0.198 \][/tex]
### Row 7: Calculating the Result (Polencia)
Given:
[tex]\[ 6^{-7} = \text{Result (Polencia)} \][/tex]
To find the result:
[tex]\[ \text{Result (Polencia)} = 6^{-7} \][/tex]
From the result:
[tex]\[ \text{Result (Polencia)} \approx 3.572 \times 10^{-6} \][/tex]
Here is the complete table with all the needed values filled in:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Base} & \text{Exponent} & \text{Polencia} \\
\hline
70 & \approx 0.458 & 7 \\
\hline
\approx 3.054 & 4 & 87 \\
\hline
4 & 2 & 16 \\
\hline
-3 & 2 & 9 \\
\hline
-2 & 3 & -8 \\
\hline
\frac{2}{3} & 4 & \approx 0.198 \\
\hline
6 & -7 & \approx 3.572 \times 10^{-6} \\
\hline
\end{array}
\][/tex]
