1. **Problem:** Given $3^s = \sqrt{3} \times 3\sqrt{7} \sqrt{9}$, find the value of $(13 + 24x)^4$. 2. **Step 1: Simplify the right side of the equation for $3^s$** - Recall that $\sqrt{3} = 3^{1/2}$. - $3\sqrt{7}$ means $3^{\sqrt{7}}$, but since this is ambiguous, assume it means $3^{1/\sqrt{7}}$ or $3^{\frac{1}{\sqrt{7}}}$. - $\sqrt{9} = 3^{1/2 \times 2} = 3^{1}$ since $\sqrt{9} = 3$. 3. **Step 2: Express all terms as powers of 3** - $3^s = 3^{1/2} \times 3^{1/\sqrt{7}} \times 3^{1}$ - Using the property $a^m \times a^n = a^{m+n}$, we get: $$3^s = 3^{1/2 + 1/\sqrt{7} + 1}$$ 4. **Step 3: Equate exponents** - Since bases are equal, exponents must be equal: $$s = \frac{1}{2} + \frac{1}{\sqrt{7}} + 1 = \frac{3}{2} + \frac{1}{\sqrt{7}}$$ 5. **Step 4: The problem asks for $(13 + 24x)^4$ but $x$ is not defined in the problem statement.** - Without a value or relation for $x$, we cannot compute $(13 + 24x)^4$. **Final answer:** Cannot determine $(13 + 24x)^4$ without additional information about $x$.