Let's consider the given table of the powers of 8:
[tex]\[
\begin{tabular}{|c|}
\hline Powers of 8 \\
\hline [tex]$8^3=512$[/tex] \\
\hline [tex]$8^2=64$[/tex] \\
\hline [tex]$8^1=8$[/tex] \\
\hline [tex]$8^0=a$[/tex] \\
\hline [tex]$8^{-1}=b$[/tex] \\
\hline [tex]$8^{-2}=c$[/tex] \\
\hline
\end{tabular}
\][/tex]
### Pattern as the Exponents Decrease:
To identify the pattern, let’s look at how the values change as the exponents decrease by 1:
- Starting from \(8^3 = 512\):
- Going from \(8^3\) to \(8^2\):
[tex]\[
512 \div 8 = 64
\][/tex]
- Going from \(8^2\) to \(8^1\):
[tex]\[
64 \div 8 = 8
\][/tex]
We notice that each time the exponent decreases by 1, the value is divided by 8.
Following this pattern:
- For \(8^0\), we divide \(8^1\) by 8:
[tex]\[
8^0 = \frac{8}{8} = 1
\][/tex]
- For \(8^{-1}\), we divide \(8^0\) by 8:
[tex]\[
8^{-1} = \frac{1}{8} = 0.125
\][/tex]
- For \(8^{-2}\), we divide \(8^{-1}\) by 8:
[tex]\[
8^{-2} = \frac{0.125}{8} = 0.015625
\][/tex]
### Values of Each Variable:
Based on the identified pattern, we can now determine the values of \(a\), \(b\), and \(c\):
- \(a = 8^0 = 1\)
- \(b = 8^{-1} = 0.125\)
- \(c = 8^{-2} = 0.015625\)
So, the values of each variable are:
[tex]\[
a = 1
\][/tex]
[tex]\[
b = 0.125
\][/tex]
[tex]\[
c = 0.015625
\][/tex]
