Question

Apply the properties of integer exponents to generate equivilant expressions to 4 to the power of zero times 5 to the power of 5 to the power of -2 / (-2) to the power of -3 with only positive exponents, then solve the expression. Your answer will be a simpified fraction with no remaining exponents.

Answer 1

We start with the expression:

\[
4^0 \times 5^{5^{-2}} \div (-2)^{-3}
\]

First, we know that any non-zero number raised to the power of zero is equal to 1:

\[
4^0 = 1
\]

So the expression simplifies to:

\[
1 \times 5^{5^{-2}} \div (-2)^{-3} = 5^{5^{-2}} \div (-2)^{-3}
\]

Now, let's simplify \(5^{5^{-2}}\). The exponent \(5^{-2}\) can be expressed as:

\[
5^{-2} = \frac{1}{5^2} = \frac{1}{25}
\]

Thus:

\[
5^{5^{-2}} = 5^{\frac{1}{25}}
\]

Now, deal with \((-2)^{-3}\):

Using the property of negative exponents, we have:

\[
(-2)^{-3} = \frac{1}{(-2)^3} = \frac{1}{-8} = -\frac{1}{8}
\]

Now, we can rewrite the whole expression as:

\[
\frac{5^{\frac{1}{25}}}{-\frac{1}{8}} = 5^{\frac{1}{25}} \times -8
\]

However, since we don't want negative terms in the final expression, we can change the sign:

\[
-8 \times 5^{\frac{1}{25}} = -\frac{8}{5^{\frac{1}{25}}}
\]

To express \(5^{\frac{1}{25}}\) with a common denominator, we can use the property of fractions:

\[
-8 \times 5^{\frac{1}{25}} = -\frac{8 \cdot 5^{\frac{1}{25}}}{1}
\]

However, since it is required to keep only positive exponents, we rearrange this:

\[
-8 \cdot 5^{\frac{1}{25}} = -\frac{8}{\sqrt[25]{5}}
\]

The final expression is:

\[
-\frac{8}{5^{\frac{1}{25}}}
\]

For complete clarity, if you want a rational expression with positive exponents only, this result will remain with its form but note that \(5^{\frac{1}{25}}\) is positive.

Thus, the simplified expression with the correspondence to fractions can be packed into:

\[
-\frac{8 \sqrt[25]{1}}{\sqrt[25]{5}} \text{ or } -\frac{8}{5^{\frac{1}{25}}}
\]

The final answer is:

\[
\text{As a simplified fraction: } -\frac{8 \sqrt[25]{1}}{5^{\frac{1}{25}}}, \text{ retaining no remaining exponents other than the } \sqrt[25]{5}.
\]

If you need a numerical value or anything else specific, let me know!