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\left(\left(-5\right)^{-2}\right)^{-4}
Use the rules of exponents to simplify the expression.
\left(-5\right)^{-2\left(-4\right)}
To raise a power to another power, multiply the exponents.
\left(-5\right)^{8}
Multiply -2 times -4.
390625
Raise -5 to the power 8.