Apply all relevant properties of exponents to simplify the following expressions. Enter your answer as an integer or completely simplified fraction without exponents. a) \( (-8)^{-2} \) Answer b) \( \left(-\frac{1}{8}\right)^{-2} \) Answer c) \( -8^{-2} \) Answer d) \( -\left(\frac{1}{8}\right)^{-2} \) Answer

Sure! Let's simplify each expression step by step. a) \( (-8)^{-2} \): Using the property of exponents that states \( a^{-n} = \frac{1}{a^n} \): \[ (-8)^{-2} = \frac{1}{(-8)^2} = \frac{1}{64} = \frac{1}{64} \] b) \( \left(-\frac{1}{8}\right)^{-2} \): Again applying the property of negative exponents: \[ \left(-\frac{1}{8}\right)^{-2} = \frac{1}{\left(-\frac{1}{8}\right)^2} = \frac{1}{\frac{1}{64}} = 64 \] c) \( -8^{-2} \): Using the exponent rule: \[ -8^{-2} = -\frac{1}{8^2} = -\frac{1}{64} \] d) \( -\left(\frac{1}{8}\right)^{-2} \): Applying the same rule for negative exponents: \[ -\left(\frac{1}{8}\right)^{-2} = -\frac{1}{\left(\frac{1}{8}\right)^2} = -\frac{1}{\frac{1}{64}} = -64 \] So the simplified answers are: a) \( \frac{1}{64} \) b) \( 64 \) c) \( -\frac{1}{64} \) d) \( -64 \)