Rationalize Denominator 64Febe

1. **State the problem:** Rationalize the denominator of the expression $$\frac{9}{4} \div (3\sqrt{5} + 6)$$. 2. **Rewrite the expression:** Division by a number is the same as multiplication by its reciprocal, so $$\frac{9}{4} \div (3\sqrt{5} + 6) = \frac{9}{4} \times \frac{1}{3\sqrt{5} + 6} = \frac{9}{4(3\sqrt{5} + 6)}$$ 3. **Rationalize the denominator:** To rationalize the denominator $$3\sqrt{5} + 6$$, multiply numerator and denominator by the conjugate $$3\sqrt{5} - 6$$: $$\frac{9}{4(3\sqrt{5} + 6)} \times \frac{3\sqrt{5} - 6}{3\sqrt{5} - 6} = \frac{9(3\sqrt{5} - 6)}{4[(3\sqrt{5} + 6)(3\sqrt{5} - 6)]}$$ 4. **Simplify the denominator using difference of squares:** $$(3\sqrt{5})^2 - 6^2 = 9 \times 5 - 36 = 45 - 36 = 9$$ So denominator becomes $$4 \times 9 = 36$$. 5. **Simplify the numerator:** $$9(3\sqrt{5} - 6) = 27\sqrt{5} - 54$$ 6. **Write the simplified expression:** $$\frac{27\sqrt{5} - 54}{36}$$ 7. **Simplify the fraction by dividing numerator and denominator by 9:** $$\frac{\cancel{9}(3\sqrt{5} - 6)}{\cancel{9}4} = \frac{3\sqrt{5} - 6}{4}$$ **Final answer:** $$\frac{3\sqrt{5} - 6}{4}$$