4. a. \( 2^{-2}+2^{-3} \quad \) b. \( 4^{0}-4^{-1} \quad \) c. \( \quad 3^{-2}+\frac{1}{4^{-2}}+15^{0}-10^{1} \)

Calculate the value by following steps: - step0: Calculate: \(2^{-2}+2^{-3}\) - step1: Evaluate the power: \(\frac{1}{4}+2^{-3}\) - step2: Evaluate the power: \(\frac{1}{4}+\frac{1}{8}\) - step3: Reduce fractions to a common denominator: \(\frac{2}{4\times 2}+\frac{1}{8}\) - step4: Multiply the numbers: \(\frac{2}{8}+\frac{1}{8}\) - step5: Transform the expression: \(\frac{2+1}{8}\) - step6: Add the numbers: \(\frac{3}{8}\) Calculate or simplify the expression \( 3^(-2) + 1/(4^(-2)) + 15^(0) - 10^(1) \). Calculate the value by following steps: - step0: Calculate: \(3^{-2}+\frac{1}{\left(4^{-2}\right)}+15^{0}-10^{1}\) - step1: Evaluate: \(3^{-2}+\frac{1}{4^{-2}}+15^{0}-10^{1}\) - step2: Evaluate the power: \(3^{-2}+\frac{1}{4^{-2}}+1-10^{1}\) - step3: Divide the terms: \(3^{-2}+16+1-10^{1}\) - step4: Evaluate the power: \(3^{-2}+16+1-10\) - step5: Evaluate the power: \(\frac{1}{9}+16+1-10\) - step6: Calculate: \(\frac{1}{9}+7\) - step7: Reduce fractions to a common denominator: \(\frac{1}{9}+\frac{7\times 9}{9}\) - step8: Transform the expression: \(\frac{1+7\times 9}{9}\) - step9: Multiply the numbers: \(\frac{1+63}{9}\) - step10: Add the numbers: \(\frac{64}{9}\) Calculate or simplify the expression \( 4^(0) - 4^(-1) \). Calculate the value by following steps: - step0: Calculate: \(4^{0}-4^{-1}\) - step1: Evaluate the power: \(1-4^{-1}\) - step2: Calculate: \(1-\frac{1}{4}\) - step3: Reduce fractions to a common denominator: \(\frac{4}{4}-\frac{1}{4}\) - step4: Transform the expression: \(\frac{4-1}{4}\) - step5: Subtract the numbers: \(\frac{3}{4}\) Let's solve each part step by step. ### Part a: \( 2^{-2} + 2^{-3} \) 1. Calculate \( 2^{-2} \): \[ 2^{-2} = \frac{1}{2^2} = \frac{1}{4} \] 2. Calculate \( 2^{-3} \): \[ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \] 3. Now, add the two results: \[ 2^{-2} + 2^{-3} = \frac{1}{4} + \frac{1}{8} \] To add these fractions, we need a common denominator, which is 8: \[ \frac{1}{4} = \frac{2}{8} \] Therefore: \[ \frac{2}{8} + \frac{1}{8} = \frac{3}{8} \] ### Result for Part a: \[ \frac{3}{8} \quad \text{or} \quad 0.375 \] --- ### Part b: \( 4^{0} - 4^{-1} \) 1. Calculate \( 4^{0} \): \[ 4^{0} = 1 \] 2. Calculate \( 4^{-1} \): \[ 4^{-1} = \frac{1}{4} \] 3. Now, subtract the two results: \[ 4^{0} - 4^{-1} = 1 - \frac{1}{4} \] To perform the subtraction: \[ 1 = \frac{4}{4} \] Therefore: \[ \frac{4}{4} - \frac{1}{4} = \frac{3}{4} \] ### Result for Part b: \[ \frac{3}{4} \quad \text{or} \quad 0.75 \] --- ### Part c: \( 3^{-2} + \frac{1}{4^{-2}} + 15^{0} - 10^{1} \) 1. Calculate \( 3^{-2} \): \[ 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \] 2. Calculate \( \frac{1}{4^{-2}} \): \[ 4^{-2} = \frac{1}{4^2} = \frac{1}{16} \quad \Rightarrow \quad \frac{1}{4^{-2}} = 16 \] 3. Calculate \( 15^{0} \): \[ 15^{0} = 1 \] 4. Calculate \( 10^{1} \): \[ 10^{1} = 10 \] 5. Now, combine all the results: \[ 3^{-2} + \frac{1}{4^{-2}} + 15^{0} - 10^{1} = \frac{1}{9} + 16 + 1 - 10 \] Simplifying this: \[ = \frac{1}{9} + 7 \] Converting 7 to a fraction with a denominator of 9: \[ 7 = \frac{63}{9} \] Therefore: \[ \frac{1}{9} + \frac{63}{9} = \frac{64}{9} \] ### Result for Part c: \[ \frac{64}{9} \quad \text{or} \quad 7\frac{1}{9} \quad \text{or} \quad 7.\dot{1} \] --- ### Summary of Results: - Part a: \( \frac{3}{8} \) or \( 0.375 \) - Part b: \( \frac{3}{4} \) or \( 0.75 \) - Part c: \( \frac{64}{9} \) or \( 7\frac{1}{9} \) or \( 7.\dot{1} \)