1. Indices: a) Simplify: \( \left(2^{3} \times 2^{4}\right) \div 2^{\text {}} \) b) Express \( 81^{1 / 4} \) in its simplest form.

Calculate the value by following steps: - step0: Calculate: \(81^{\frac{1}{4}}\) - step1: Factor the number: \(\left(3^{4}\right)^{\frac{1}{4}}\) - step2: Evaluate the power: \(3\) Calculate or simplify the expression \( (2^3 * 2^4) / 2 \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(2^{3}\times 2^{4}\right)}{2}\) - step1: Remove the parentheses: \(\frac{2^{3}\times 2^{4}}{2}\) - step2: Multiply: \(\frac{2^{7}}{2}\) - step3: Divide the numbers: \(2^{7-1}\) - step4: Subtract the terms: \(2^{6}\) - step5: Evaluate the power: \(64\) Let's simplify each part step by step: ### a) Simplify \( \left(2^{3} \times 2^{4}\right) \div 2 \) 1. **Combine the powers of 2**: \[ 2^{3} \times 2^{4} = 2^{3+4} = 2^{7} \] 2. **Now divide by \( 2 \)**: \[ 2^{7} \div 2 = 2^{7-1} = 2^{6} \] 3. **Calculate \( 2^{6} \)**: \[ 2^{6} = 64 \] So, the simplified result is \( 64 \). ### b) Express \( 81^{1 / 4} \) in its simplest form. 1. **Recognize that \( 81 \) can be expressed as a power of \( 3 \)**: \[ 81 = 3^{4} \] 2. **Now apply the exponent**: \[ 81^{1/4} = (3^{4})^{1/4} = 3^{4 \cdot (1/4)} = 3^{1} = 3 \] So, the simplest form of \( 81^{1 / 4} \) is \( 3 \). ### Final Answers: - a) \( 64 \) - b) \( 3 \)