To simplify the expression \(\left[(-5)^{-7}(-5)^8\right]^3\), follow these steps:
1. Apply the product of powers property:
[tex]$[tex]$ \left[(-5)^{-7}(-5)^8\right]^3 $[/tex]$[/tex]
Using the property of exponents \((a^m \cdot a^n = a^{m+n})\), combine the exponents inside the brackets:
[tex]$[tex]$ (-5)^{-7+8} = (-5)^1 $[/tex]$[/tex]
2. Simplify inside the brackets:
[tex]$[tex]$ (-5)^1 = -5 $[/tex]$[/tex]
3. Apply the power of a power property:
Next, use the property \((a^m)^n = a^{m \cdot n}\) to simplify the expression \((-5)^1\) raised to the power of 3:
[tex]$[tex]$ \left[(-5)^1\right]^3 = (-5)^{1 \cdot 3} = (-5)^3 $[/tex]$[/tex]
4. Calculate the final result:
Now raise \(-5\) to the power of \(3\):
[tex]$[tex]$ (-5)^3 = -125 $[/tex]$[/tex]
Therefore, the value of the expression \(\left[(-5)^{-7}(-5)^8\right]^3\) is:
[tex]$[tex]$ \boxed{-125} $[/tex]$[/tex]
