Exponent Evaluation 9Ff75A

1. **Problem 46:** Evaluate $0.64^{\frac{7}{2}}$ and $0.64^5$. 2. **Step 1:** Express $0.64$ as a fraction: $0.64 = \frac{64}{100} = \frac{16}{25}$. 3. **Step 2:** Calculate $0.64^{\frac{7}{2}} = \left(\frac{16}{25}\right)^{\frac{7}{2}} = \left(\left(\frac{16}{25}\right)^{\frac{1}{2}}\right)^7$. 4. **Step 3:** Find the square root: $\left(\frac{16}{25}\right)^{\frac{1}{2}} = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5}$. 5. **Step 4:** Raise to the 7th power: $\left(\frac{4}{5}\right)^7 = \frac{4^7}{5^7} = \frac{16384}{78125}$. 6. **Step 5:** Approximate decimal value: $\frac{16384}{78125} \approx 0.2097$. 7. **Step 6:** Calculate $0.64^5 = \left(\frac{16}{25}\right)^5 = \frac{16^5}{25^5} = \frac{1048576}{9765625} \approx 0.1074$. --- 8. **Problem 47:** Simplify $\left(\frac{5}{2} a^{-4} b^{7}\right)^{-3}$. 9. **Step 1:** Apply the negative exponent: $\left(\frac{5}{2}\right)^{-3} a^{(-4)(-3)} b^{7(-3)} = \left(\frac{2}{5}\right)^3 a^{12} b^{-21}$. 10. **Step 2:** Calculate $\left(\frac{2}{5}\right)^3 = \frac{2^3}{5^3} = \frac{8}{125}$. 11. **Step 3:** Write the expression: $\frac{8}{125} a^{12} b^{-21} = \frac{8 a^{12}}{125 b^{21}}$. --- 12. **Problem 48:** Evaluate $(a^{-4} b^{-3})(a^{3} b^{-4})$ for $a = -1$ and $b = 3$. 13. **Step 1:** Combine like bases: $a^{-4 + 3} b^{-3 + (-4)} = a^{-1} b^{-7} = \frac{1}{a} \cdot \frac{1}{b^7} = \frac{1}{a b^7}$. 14. **Step 2:** Substitute values: $a = -1$, $b = 3$. 15. **Step 3:** Calculate denominator: $a b^7 = (-1) \times 3^7 = (-1) \times 2187 = -2187$. 16. **Step 4:** Final value: $\frac{1}{-2187} = -\frac{1}{2187}$. --- **Final answers:** - For problem 46: $0.64^{\frac{7}{2}} = \frac{16384}{78125} \approx 0.2097$, $0.64^5 = \frac{1048576}{9765625} \approx 0.1074$. - For problem 47: Simplified form is $\frac{8 a^{12}}{125 b^{21}}$ (option b). - For problem 48: Value is $-\frac{1}{2187}$ (option a).