Let's simplify the given expression step-by-step:

$\left(\frac{5^{-2} \times 5^3}{5^5}\right)^0$

Step 1: Simplify the powers in the numerator

Use the property of exponents $a^m \times a^n = a^{m+n}$:

$5^{-2} \times 5^3 = 5^{-2 + 3} = 5^1 = 5$

Step 2: Substitute back into the expression

$\left(\frac{5}{5^5}\right)^0$

Step 3: Simplify the fraction using the property $\frac{a^m}{a^n} = a^{m-n}$

$\frac{5}{5^5} = 5^{1 - 5} = 5^{-4}$

Step 4: The expression now becomes

$(5^{-4})^0$

Step 5: Apply the rule $a^0 = 1$ (for any non-zero $a$)

$(5^{-4})^0 = 1$

Final Answer

$\boxed{1}$

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Do you need further explanation or have any questions?

Related Questions:

1. What is the general rule for simplifying expressions with powers of the same base?
2. How do you simplify $(a^m)^n$?
3. What happens if an exponent is negative?
4. What is the rule for raising a fraction to the power of zero?
5. How do you simplify $a^0$ when $a \neq 0$?

Tip:

Remember, any non-zero number raised to the power of zero is always equal to 1.