Sure! Let's analyze the given table to understand the pattern in the values as the exponents increase.
[tex]\[
\begin{tabular}{|c|c|}
\hline
\text{Powers of 2} & \text{Value} \\
\hline
2^{-5} & \frac{1}{32} \\
\hline
2^{-4} & \frac{1}{16} \\
\hline
2^{-3} & \frac{1}{8} \\
\hline
2^{-2} & \frac{1}{4} \\
\hline
2^{-1} & \frac{1}{2} \\
\hline
2^0 & 1 \\
\hline
\end{tabular}
\][/tex]
1. Start with Exponent -5:
- For [tex]\(2^{-5}\)[/tex], the value is [tex]\(\frac{1}{32}\)[/tex], which can also be written as 0.03125.
2. Exponent -4:
- For [tex]\(2^{-4}\)[/tex], the value is [tex]\(\frac{1}{16}\)[/tex] or 0.0625.
- Notice that [tex]\(0.0625 = 2 \times 0.03125\)[/tex]. This means that each subsequent value is double the previous value.
3. Exponent -3:
- For [tex]\(2^{-3}\)[/tex], the value is [tex]\(\frac{1}{8}\)[/tex] or 0.125.
- Similarly, [tex]\(0.125 = 2 \times 0.0625\)[/tex].
4. Exponent -2:
- For [tex]\(2^{-2}\)[/tex], the value is [tex]\(\frac{1}{4}\)[/tex] or 0.25.
- Likewise, [tex]\(0.25 = 2 \times 0.125\)[/tex].
5. Exponent -1:
- For [tex]\(2^{-1}\)[/tex], the value is [tex]\(\frac{1}{2}\)[/tex] or 0.5.
- Here, [tex]\(0.5 = 2 \times 0.25\)[/tex].
6. Exponent 0:
- For [tex]\(2^0\)[/tex], the value is 1.
- Consequently, [tex]\(1 = 2 \times 0.5\)[/tex].
### Observing the Pattern:
- Each value in the table is exactly twice the previous value.
- This makes sense mathematically because increasing the exponent by 1 for base 2 doubles the result.
To summarize:
- For each increase in the exponent by 1 in [tex]\(2^n\)[/tex], the corresponding value doubles.
Thus, the pattern in the values as the exponents increase is that each subsequent value is double the previous value.
