3. a. \( 2^{5} \times 2^{-3} \) b. \( 4^{3} \div 4^{-4} \) c. \( 2 a^{-2} \times 3 a^{3} \times a^{0} \)

Simplify the expression by following steps: - step0: Solution: \(2a^{-2}\times 3a^{3}\times a^{0}\) - step1: Evaluate the power: \(2a^{-2}\times 3a^{3}\times 1\) - step2: Rewrite the expression: \(2a^{-2}\times 3a^{3}\) - step3: Multiply the terms: \(6a^{-2}\times a^{3}\) - step4: Multiply the terms: \(6a^{-2+3}\) - step5: Add the numbers: \(6a\) Calculate or simplify the expression \( 4^{3} / 4^{-4} \). Calculate the value by following steps: - step0: Calculate: \(\frac{4^{3}}{4^{-4}}\) - step1: Multiply by \(a^{-n}:\) \(4^{3}\times 4^{4}\) - step2: Multiply: \(4^{3+4}\) - step3: Add the numbers: \(4^{7}\) - step4: Evaluate the power: \(16384\) Calculate or simplify the expression \( 2^{5} * 2^{-3} \). Calculate the value by following steps: - step0: Calculate: \(2^{5}\times 2^{-3}\) - step1: Multiply: \(2^{5-3}\) - step2: Subtract the numbers: \(2^{2}\) - step3: Evaluate the power: \(4\) Here are the solutions for each part of the problem: ### a. \( 2^{5} \times 2^{-3} \) To simplify this expression, we can use the property of exponents that states \( a^{m} \times a^{n} = a^{m+n} \). \[ 2^{5} \times 2^{-3} = 2^{5 + (-3)} = 2^{2} = 4 \] **Result:** \( 4 \) --- ### b. \( 4^{3} \div 4^{-4} \) For division of exponents, we use the property \( a^{m} \div a^{n} = a^{m-n} \). \[ 4^{3} \div 4^{-4} = 4^{3 - (-4)} = 4^{3 + 4} = 4^{7} \] Calculating \( 4^{7} \): \[ 4^{7} = (2^{2})^{7} = 2^{14} = 16384 \] **Result:** \( 16384 \) --- ### c. \( 2 a^{-2} \times 3 a^{3} \times a^{0} \) Using the properties of exponents, we can simplify this expression step by step: 1. Combine the coefficients: \( 2 \times 3 = 6 \) 2. Combine the \( a \) terms: \( a^{-2} \times a^{3} \times a^{0} = a^{-2 + 3 + 0} = a^{1} = a \) Putting it all together: \[ 2 a^{-2} \times 3 a^{3} \times a^{0} = 6 a \] **Result:** \( 6a \) --- ### Summary of Results: - a. \( 4 \) - b. \( 16384 \) - c. \( 6a \)