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The image asks to compare the values of \(5^{8}\) and \(10^{6}\) by writing them in the same base with an index of 2.

To solve this problem, we will convert both \(5^8\) and \(10^6\) to base 2. We can use the logarithm base 2 to do this.

### Step-by-Step Solution

1. **Convert \(5^8\) to base 2:**

   \[
   5^8 = (2^{\log_2(5)})^8 = 2^{8 \cdot \log_2(5)}
   \]

2. **Convert \(10^6\) to base 2:**

   \[
   10^6 = (2^{\log_2(10)})^6 = 2^{6 \cdot \log_2(10)}
   \]

3. **Calculate the logarithms:**

   Using the approximate values:
   \[
   \log_2(5) \approx 2.32, \quad \log_2(10) = \log_2(2 \cdot 5) = \log_2(2) + \log_2(5) = 1 + 2.32 = 3.32
   \]

4. **Substitute the logarithms:**

   - For \(5^8\):

   \[
   5^8 \approx 2^{8 \cdot 2.32} = 2^{18.56}
   \]

   - For \(10^6\):

   \[
   10^6 \approx 2^{6 \cdot 3.32} = 2^{19.92}
   \]

5. **Compare the values:**

   Since \(2^{18.56} < 2^{19.92}\), we have:

   \[
   5^8 < 10^6
   \]

### Conclusion

\(10^6\) is greater than \(5^8\) when both are written in base 2. 

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Compare the value of 5^8 and 10^6. Write them in the same base with index 2.

Understand how to compare the values of 5^8 and 10^6 by converting both to the same base, with detailed step-by-step instructions. Suitable for high school students in Grades 9-12.

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