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

To simplify the given expressions using properties of exponents: a) \( (-8)^{-2} \) When simplifying this, remember that a negative exponent indicates the reciprocal, so: \[ (-8)^{-2} = \frac{1}{(-8)^2} = \frac{1}{64} \] **Answer:** \( \frac{1}{64} \) b) \( \left(-\frac{1}{8}\right)^{-2} \) Again, apply the negative exponent rule: \[ \left(-\frac{1}{8}\right)^{-2} = \frac{1}{\left(-\frac{1}{8}\right)^2} = \frac{1}{\frac{1}{64}} = 64 \] **Answer:** \( 64 \) c) \( -8^{-2} \) For this expression, first deal with the exponent: \[ -8^{-2} = -\frac{1}{8^2} = -\frac{1}{64} \] **Answer:** \( -\frac{1}{64} \) d) \( -\left(\frac{1}{8}\right)^{-2} \) Use the negative exponent rule here too: \[ -\left(\frac{1}{8}\right)^{-2} = -\frac{1}{\left(\frac{1}{8}\right)^2} = -\frac{1}{\frac{1}{64}} = -64 \] **Answer:** \( -64 \)