Question

equations that are equivalent to 5^5 x 8^(-7)

Answer 1

To find equations that are equivalent to \( 5^5 \times 8^{-7} \), we can first simplify or rewrite the terms involved.

1. Rewrite \( 8 \) in terms of base \( 2 \):
\[
8 = 2^3
\]
Therefore, \( 8^{-7} = (2^3)^{-7} = 2^{-21} \).

Now we can express our original equation:
\[
5^5 \times 8^{-7} = 5^5 \times 2^{-21}
\]

2. Combine the bases and express as a fraction:
\[
5^5 \times 2^{-21} = \frac{5^5}{2^{21}}
\]

3. You can also express this in terms of a single fraction:
\[
5^5 \times 8^{-7} = \frac{5^5}{8^7}
\]

4. Using the original form, you can extend it out in different equivalent ways. For instance, using properties of exponents or logarithms, like writing it in logarithmic form:
\[
\log(5^5 \times 8^{-7}) = 5 \log(5) - 7 \log(8)
\]

5. Express it in terms of a different base:
\[
= (5 \times 8^{-7/5})^5
\]

### Summary of Equivalent Forms
- \( 5^5 \times 2^{-21} \)
- \( \frac{5^5}{2^{21}} \)
- \( \frac{5^5}{8^7} \)
- \( \log(5^5 \times 8^{-7}) = 5 \log(5) - 7 \log(8) \)
- \( (5 \times 8^{-7/5})^5 \)

All these forms are equivalent to \( 5^5 \times 8^{-7} \).