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

$\left[ (-2.5)^{7^2} : (-2.5)^{-8^{-2}} \right] - \left( \frac{5}{2} : 2.5 \right)^0$

Step 1: Simplify the exponents

• $7^2 = 49$, so the first term becomes $(-2.5)^{49}$.
• $-8^{-2} = -\frac{1}{64}$, so the second term becomes $(-2.5)^{-\frac{1}{64}}$.

Now the expression looks like this:

$\left[ (-2.5)^{49} : (-2.5)^{-\frac{1}{64}} \right] - \left( \frac{5}{2} : 2.5 \right)^0$

Step 2: Apply exponent rules

Using the rule $a^m : a^n = a^{m-n}$:

$(-2.5)^{49 - \left(-\frac{1}{64}\right)} = (-2.5)^{49 + \frac{1}{64}}$

Step 3: Simplify the second part

$\frac{5}{2} : 2.5 = \frac{5/2}{5/2} = 1$ And any number to the power of 0 is 1: $1^0 = 1$

Final Expression:

$(-2.5)^{49 + \frac{1}{64}} - 1$

This is the simplified form of the expression. If you'd like, I can compute the numerical value as well. Let me know if you'd like further details or clarification!

1. Would you like me to compute the numerical value of $(-2.5)^{49 + \frac{1}{64}}$?
2. Do you want a deeper explanation on exponent rules?
3. Should I explain negative and fractional exponents in more detail?
4. Would you like to see how to simplify similar complex expressions?
5. Do you want to explore how the base being negative affects the exponentiation?

Tip: Remember, any non-zero number raised to the power of zero is always 1!