1. **State the problem:** Simplify the expression $$\frac{24\sqrt{225x^{22}}}{4\sqrt{5x^4}}$$. 2. **Recall the rule for simplifying square roots:** $$\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$$ and $$\sqrt{x^{2n}} = x^n$$ for even powers. 3. **Simplify inside the square roots:** $$\sqrt{225x^{22}} = \sqrt{225} \cdot \sqrt{x^{22}} = 15 \cdot x^{11}$$ $$\sqrt{5x^4} = \sqrt{5} \cdot \sqrt{x^4} = \sqrt{5} \cdot x^2$$ 4. **Substitute back into the expression:** $$\frac{24 \cdot 15 x^{11}}{4 \cdot \sqrt{5} x^2} = \frac{360 x^{11}}{4 x^2 \sqrt{5}}$$ 5. **Simplify the fraction by dividing numerator and denominator by 4:** $$\frac{\cancel{360}^{90} x^{11}}{\cancel{4}^1 x^2 \sqrt{5}} = \frac{90 x^{11}}{x^2 \sqrt{5}}$$ 6. **Simplify the powers of $x$ by subtracting exponents:** $$\frac{90 x^{\cancel{11}^{9}}}{x^{\cancel{2}^0} \sqrt{5}} = \frac{90 x^9}{\sqrt{5}}$$ 7. **Rationalize the denominator:** $$\frac{90 x^9}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{90 x^9 \sqrt{5}}{5}$$ 8. **Simplify the fraction:** $$\frac{\cancel{90}^{18} x^9 \sqrt{5}}{\cancel{5}^1} = 18 x^9 \sqrt{5}$$ **Final answer:** $$18 x^9 \sqrt{5}$$