Let's simplify the given expression step by step.
Starting with the original expression:
\[ [(–5)^{-7} (–5)^{8}]^{3} \]
Step 1: Apply the product of powers.
When multiplying two exponents with the same base, we add the exponents:
\[ (-5)^{-7} \cdot (-5)^{8} = (-5)^{-7 + 8} = (-5)^{1} \]
Now, we can rewrite the expression as:
\[ [(-5)^{1}]^{3} \]
Step 2: Apply the power of a power.
When raising a power to another power, we multiply the exponents:
\[ [(-5)^{1}]^{3} = (-5)^{1 \cdot 3} = (-5)^{3} \]
Step 3: Simplify.
Now we calculate \((-5)^3\):
\[ (-5)^3 = -125 \]
Final Result:
The value of the expression is \(-125\).