Let's break down the expression step by step:

$\left( 3^0 + 7^{-1} - 8^0 \right)^{-1}$

Step 1: Simplify individual terms inside the parentheses.

1. $3^0 = 1$ (Any non-zero number to the power of 0 is 1).
2. $7^{-1} = \frac{1}{7}$ (Any number raised to the power of -1 is the reciprocal of that number).
3. $8^0 = 1$ (Again, any non-zero number to the power of 0 is 1).

Now substitute these values into the expression:

$\left( 1 + \frac{1}{7} - 1 \right)^{-1}$

Step 2: Simplify the expression inside the parentheses.

$1 + \frac{1}{7} - 1 = \frac{1}{7}$

Step 3: Apply the exponent of -1 to the result.

$\left( \frac{1}{7} \right)^{-1} = 7$

Final Answer:

$7$

Would you like more details or have any further questions?

Here are 5 related questions for further exploration:

1. What is $(2^0 + 5^{-1} - 6^0)^{-1}$?
2. How do you simplify expressions with negative exponents?
3. Why is any number to the power of zero equal to 1?
4. How would the answer change if the base numbers were different, such as $3^2$ instead of $3^0$?
5. What are the properties of exponents and how do they apply in different mathematical contexts?

Tip: Whenever you encounter a negative exponent, remember that it just means taking the reciprocal of the base raised to the positive exponent.