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!
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