8. \( \left(\frac{36}{64}\right)^{-\frac{3}{2}} \quad \) 9. \( (2,25)^{-\frac{1}{2}} \)

Let's solve each expression step by step. --- ### **8. \( \left(\frac{36}{64}\right)^{-\frac{3}{2}} \)** **Step 1:** Simplify the fraction. \[ \frac{36}{64} = \frac{9}{16} \quad \text{(both numerator and denominator are divisible by 4)} \] **Step 2:** Rewrite the expression using the simplified fraction. \[ \left(\frac{9}{16}\right)^{-\frac{3}{2}} \] **Step 3:** Apply the negative exponent rule \( a^{-b} = \frac{1}{a^b} \). \[ \left(\frac{9}{16}\right)^{-\frac{3}{2}} = \left(\frac{16}{9}\right)^{\frac{3}{2}} \] **Step 4:** Simplify the exponent. \[ \left(\frac{16}{9}\right)^{\frac{3}{2}} = \left(\sqrt{\frac{16}{9}}\right)^3 = \left(\frac{4}{3}\right)^3 = \frac{64}{27} \] **Final Answer:** \[ \boxed{\dfrac{64}{27}} \] --- ### **9. \( 2.25^{-\frac{1}{2}} \)** **Step 1:** Recognize that \( 2.25 \) is equivalent to \( \frac{9}{4} \). \[ 2.25 = \frac{9}{4} \] **Step 2:** Rewrite the expression using the fraction. \[ \left(\frac{9}{4}\right)^{-\frac{1}{2}} \] **Step 3:** Apply the negative exponent rule \( a^{-b} = \frac{1}{a^b} \). \[ \left(\frac{9}{4}\right)^{-\frac{1}{2}} = \left(\frac{4}{9}\right)^{\frac{1}{2}} \] **Step 4:** Simplify the square root. \[ \left(\frac{4}{9}\right)^{\frac{1}{2}} = \frac{2}{3} \] **Final Answer:** \[ \boxed{\dfrac{2}{3}} \] ---