(a) \( \left[\left(\frac{8}{19}\right)^{17} \div\left(\frac{8}{19}\right)^{2}\right] \times\left(\frac{8}{19}\right)^{2} \) (b) \( \frac{3^{8} \times 3^{3} \times 5^{8} \times 5^{0}}{(3 \times 5)^{5}} \)

Calculate the value by following steps: - step0: Calculate: \(\left(\frac{\left(\frac{8}{19}\right)^{17}}{\left(\frac{8}{19}\right)^{2}}\right)\left(\frac{8}{19}\right)^{2}\) - step1: Multiply by \(a^{-n}:\) \(\left(\left(\frac{8}{19}\right)^{17}\left(\frac{8}{19}\right)^{-2}\right)\left(\frac{8}{19}\right)^{2}\) - step2: Multiply the numbers: \(\frac{8^{15}}{19^{15}}\times \left(\frac{8}{19}\right)^{2}\) - step3: Evaluate the power: \(\frac{8^{15}}{19^{15}}\times \frac{8^{2}}{19^{2}}\) - step4: Multiply the fractions: \(\frac{8^{15}\times 8^{2}}{19^{15}\times 19^{2}}\) - step5: Multiply: \(\frac{8^{17}}{19^{17}}\) Calculate or simplify the expression \( (3^8 * 3^3 * 5^8 * 5^0) / ((3 * 5)^5) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(3^{8}\times 3^{3}\times 5^{8}\times 5^{0}\right)}{\left(\left(3\times 5\right)^{5}\right)}\) - step1: Evaluate the power: \(\frac{\left(3^{8}\times 3^{3}\times 5^{8}\times 1\right)}{\left(\left(3\times 5\right)^{5}\right)}\) - step2: Evaluate: \(\frac{\left(3^{8}\times 3^{3}\times 5^{8}\times 1\right)}{\left(3\times 5\right)^{5}}\) - step3: Remove the parentheses: \(\frac{3^{8}\times 3^{3}\times 5^{8}\times 1}{\left(3\times 5\right)^{5}}\) - step4: Multiply the numbers: \(\frac{3^{8}\times 3^{3}\times 5^{8}\times 1}{15^{5}}\) - step5: Multiply: \(\frac{3^{11}\times 5^{8}}{15^{5}}\) - step6: Factor the expression: \(\frac{3^{11}\times 5^{8}}{3^{5}\times 5^{5}}\) - step7: Reduce the fraction: \(3^{6}\times 5^{3}\) - step8: Expand the expression: \(729\times 5^{3}\) - step9: Expand the expression: \(729\times 125\) - step10: Multiply the numbers: \(91125\) Let's solve each part step by step. ### Part (a) We need to simplify the expression: \[ \left[\left(\frac{8}{19}\right)^{17} \div\left(\frac{8}{19}\right)^{2}\right] \times\left(\frac{8}{19}\right)^{2} \] 1. **Division of Exponents**: When dividing powers with the same base, we subtract the exponents: \[ \left(\frac{8}{19}\right)^{17} \div \left(\frac{8}{19}\right)^{2} = \left(\frac{8}{19}\right)^{17 - 2} = \left(\frac{8}{19}\right)^{15} \] 2. **Multiplication of Exponents**: Now we multiply this result by \(\left(\frac{8}{19}\right)^{2}\): \[ \left(\frac{8}{19}\right)^{15} \times \left(\frac{8}{19}\right)^{2} = \left(\frac{8}{19}\right)^{15 + 2} = \left(\frac{8}{19}\right)^{17} \] Thus, the simplified result for part (a) is: \[ \left(\frac{8}{19}\right)^{17} \quad \text{or approximately } 4.108834 \times 10^{-7} \] ### Part (b) We need to simplify the expression: \[ \frac{3^{8} \times 3^{3} \times 5^{8} \times 5^{0}}{(3 \times 5)^{5}} \] 1. **Combine the Numerator**: We can combine the powers of the same base: \[ 3^{8} \times 3^{3} = 3^{8 + 3} = 3^{11} \] And since \(5^{0} = 1\), we have: \[ 5^{8} \times 5^{0} = 5^{8} \] Therefore, the numerator becomes: \[ 3^{11} \times 5^{8} \] 2. **Simplify the Denominator**: The denominator can be simplified as follows: \[ (3 \times 5)^{5} = 3^{5} \times 5^{5} \] 3. **Combine the Fraction**: Now we can write the entire expression: \[ \frac{3^{11} \times 5^{8}}{3^{5} \times 5^{5}} = \frac{3^{11}}{3^{5}} \times \frac{5^{8}}{5^{5}} = 3^{11 - 5} \times 5^{8 - 5} = 3^{6} \times 5^{3} \] 4. **Calculate the Final Result**: Now we can calculate \(3^{6} \times 5^{3}\): \[ 3^{6} = 729 \quad \text{and} \quad 5^{3} = 125 \] Therefore: \[ 3^{6} \times 5^{3} = 729 \times 125 = 91125 \] Thus, the simplified result for part (b) is: \[ 91125 \] ### Final Results - Part (a): \(\left(\frac{8}{19}\right)^{17}\) or approximately \(4.108834 \times 10^{-7}\) - Part (b): \(91125\)