Rationalize Denominator 2888Ff

1. **State the problem:** Rationalize the denominator of the expression $$\frac{4}{3\sqrt{5} + 6} \cdot \frac{3\sqrt{5} - 6}{3\sqrt{5} - 6}$$. 2. **Formula and rules:** To rationalize a denominator with a sum involving a square root, multiply numerator and denominator by the conjugate of the denominator. The conjugate of $$a + b$$ is $$a - b$$, and multiplying them results in a difference of squares: $$ (a+b)(a-b) = a^2 - b^2 $$. 3. **Apply the conjugate:** Multiply numerator and denominator by $$3\sqrt{5} - 6$$: $$\frac{4}{3\sqrt{5} + 6} \times \frac{3\sqrt{5} - 6}{3\sqrt{5} - 6} = \frac{4(3\sqrt{5} - 6)}{(3\sqrt{5} + 6)(3\sqrt{5} - 6)}$$ 4. **Calculate the denominator using difference of squares:** $$ (3\sqrt{5})^2 - 6^2 = 9 \times 5 - 36 = 45 - 36 = 9 $$ 5. **Calculate the numerator:** $$4(3\sqrt{5} - 6) = 12\sqrt{5} - 24$$ 6. **Write the fraction:** $$\frac{12\sqrt{5} - 24}{9}$$ 7. **Simplify by dividing numerator and denominator by 3:** $$\frac{\cancel{3} \times (4\sqrt{5} - 8)}{\cancel{3} \times 3} = \frac{4\sqrt{5} - 8}{3}$$ **Final answer:** $$\frac{4\sqrt{5} - 8}{3}$$