1. **State the problem:** Simplify the expression $$4^{\frac{3}{4}} \times \sqrt{\left(\frac{6}{5}\right)^{15} \times \left(\frac{6}{5}\right)^9}$$. 2. **Recall the rules:** - When multiplying powers with the same base, add the exponents: $$a^m \times a^n = a^{m+n}$$. - The square root of a product is the product of the square roots: $$\sqrt{xy} = \sqrt{x} \times \sqrt{y}$$. - The square root can be expressed as a power of $$\frac{1}{2}$$: $$\sqrt{x} = x^{\frac{1}{2}}$$. 3. **Simplify inside the square root:** $$\left(\frac{6}{5}\right)^{15} \times \left(\frac{6}{5}\right)^9 = \left(\frac{6}{5}\right)^{15+9} = \left(\frac{6}{5}\right)^{24}$$ 4. **Apply the square root as a power of $$\frac{1}{2}$$:** $$\sqrt{\left(\frac{6}{5}\right)^{24}} = \left(\frac{6}{5}\right)^{24 \times \frac{1}{2}} = \left(\frac{6}{5}\right)^{12}$$ 5. **Rewrite the original expression:** $$4^{\frac{3}{4}} \times \left(\frac{6}{5}\right)^{12}$$ 6. **Express 4 as a power of 2:** $$4 = 2^2$$, so $$4^{\frac{3}{4}} = \left(2^2\right)^{\frac{3}{4}} = 2^{2 \times \frac{3}{4}} = 2^{\frac{3}{2}}$$ 7. **Final simplified expression:** $$2^{\frac{3}{2}} \times \left(\frac{6}{5}\right)^{12}$$ **Answer:** $$2^{\frac{3}{2}} \times \left(\frac{6}{5}\right)^{12}$$