Question

Select all expressions that are equivalent to 5^-8 * 5^3
1. (5^5)^-1
2. 1/5^-24
3. 5^-24
4.5^-5

Answer 1

To simplify the original expression \(5^{-8} \cdot 5^{3}\), we can use the property of exponents that states \(a^m \cdot a^n = a^{m+n}\).

So, we have:

\[
5^{-8} \cdot 5^{3} = 5^{-8 + 3} = 5^{-5}
\]

Now, let's evaluate the given options one by one:

1. **\((5^{5})^{-1}\)**:
Using the property \((a^{m})^{-1} = a^{-m}\), we get:
\[
(5^{5})^{-1} = 5^{-5}
\]
This is equivalent to \(5^{-5}\).

2. **\(\frac{1}{5^{-24}}\)**:
This can be rewritten using the property \(\frac{1}{a^{-m}} = a^{m}\):
\[
\frac{1}{5^{-24}} = 5^{24}
\]
This is not equivalent to \(5^{-5}\).

3. **\(5^{-24}\)**:
This expression is already in the form of \(5^{m}\) and is \(5^{-24}\), which is not equivalent to \(5^{-5}\).

4. **\(5^{-5}\)**:
This is exactly what we simplified our original expression to, so it is equivalent.

From this analysis, the equivalent expressions to \(5^{-8} \cdot 5^{3}\) are:

- \( (5^{5})^{-1} \)
- \( 5^{-5} \)

Thus, the correct selections are:

1. \((5^{5})^{-1}\)
4. \(5^{-5}\)