3. \( (a-b)^{4} \) Expand and simplify the following: \( \begin{array}{llll}\text { (a) }(-2)^{3} & \text { (b) }(-2)^{4} & \text { (c) }(-2)^{3} \\ \text { (d) }-(-2)^{4} & \text { (e) }(-a)^{5} & \text { (f) }(-a)^{6} \\ \text { (g) }-(-a)^{5} & \text { (h) }-(-a)^{6} & \text { (i) }(-1)^{20} \\ \text { (j) }(-1)^{23} & \text { (k) }(-1)^{200} & \text { (l) } & (-1)^{203}\end{array} \)

Calculate the value by following steps: - step0: Calculate: \(\left(-1\right)^{203}\) - step1: Evaluate the power: \(-1\) Calculate or simplify the expression \( (-1)^{20} \). Calculate the value by following steps: - step0: Calculate: \(\left(-1\right)^{20}\) - step1: Evaluate the power: \(1\) Calculate or simplify the expression \( (-a)^6 \). Simplify the expression by following steps: - step0: Solution: \(\left(-a\right)^{6}\) - step1: Determine the sign: \(a^{6}\) Calculate or simplify the expression \( (-1)^{23} \). Calculate the value by following steps: - step0: Calculate: \(\left(-1\right)^{23}\) - step1: Evaluate the power: \(-1\) Calculate or simplify the expression \( -(-a)^5 \). Simplify the expression by following steps: - step0: Solution: \(-\left(-a\right)^{5}\) - step1: Determine the sign: \(a^{5}\) Calculate or simplify the expression \( (-2)^3 \). Calculate the value by following steps: - step0: Calculate: \(\left(-2\right)^{3}\) - step1: Evaluate the power: \(-2^{3}\) - step2: Evaluate the power: \(-8\) Calculate or simplify the expression \( (-2)^3 \). Calculate the value by following steps: - step0: Calculate: \(\left(-2\right)^{3}\) - step1: Evaluate the power: \(-2^{3}\) - step2: Evaluate the power: \(-8\) Calculate or simplify the expression \( (-2)^4 \). Calculate the value by following steps: - step0: Calculate: \(\left(-2\right)^{4}\) - step1: Determine the sign: \(2^{4}\) - step2: Evaluate the power: \(16\) Calculate or simplify the expression \( -(-2)^4 \). Calculate the value by following steps: - step0: Calculate: \(-\left(-2\right)^{4}\) - step1: Evaluate the power: \(-16\) Calculate or simplify the expression \( -(-a)^6 \). Simplify the expression by following steps: - step0: Solution: \(-\left(-a\right)^{6}\) - step1: Determine the sign: \(-a^{6}\) Calculate or simplify the expression \( (-1)^{200} \). Calculate the value by following steps: - step0: Calculate: \(\left(-1\right)^{200}\) - step1: Evaluate the power: \(1\) Calculate or simplify the expression \( (-a)^5 \). Simplify the expression by following steps: - step0: Solution: \(\left(-a\right)^{5}\) - step1: Determine the sign: \(-a^{5}\) Let's expand and simplify the expressions step by step: 1. **Calculate \((-2)^3\)**: \[ (-2)^3 = -2 \times -2 \times -2 = -8 \] 2. **Calculate \((-2)^4\)**: \[ (-2)^4 = -2 \times -2 \times -2 \times -2 = 16 \] 3. **Calculate \((-2)^3\)** again (same as step 1): \[ (-2)^3 = -8 \] 4. **Calculate \(-(-2)^4\)**: \[ -(-2)^4 = -16 \] 5. **Calculate \((-a)^5\)**: \[ (-a)^5 = -a^5 \] 6. **Calculate \((-a)^6\)**: \[ (-a)^6 = a^6 \] 7. **Calculate \(-(-a)^5\)**: \[ -(-a)^5 = a^5 \] 8. **Calculate \(-(-a)^6\)**: \[ -(-a)^6 = -a^6 \] 9. **Calculate \((-1)^{20}\)**: \[ (-1)^{20} = 1 \] 10. **Calculate \((-1)^{23}\)**: \[ (-1)^{23} = -1 \] 11. **Calculate \((-1)^{200}\)**: \[ (-1)^{200} = 1 \] 12. **Calculate \((-1)^{203}\)**: \[ (-1)^{203} = -1 \] ### Summary of Results: - (a) \((-2)^3 = -8\) - (b) \((-2)^4 = 16\) - (c) \((-2)^3 = -8\) - (d) \(-(-2)^4 = -16\) - (e) \((-a)^5 = -a^5\) - (f) \((-a)^6 = a^6\) - (g) \(-(-a)^5 = a^5\) - (h) \(-(-a)^6 = -a^6\) - (i) \((-1)^{20} = 1\) - (j) \((-1)^{23} = -1\) - (k) \((-1)^{200} = 1\) - (l) \((-1)^{203} = -1\)