Question:

**Step 1: Determine the value of $a^{(\log_3 7)^2}$** Given the equation $a^{\log_3 7} = 27$. To find $a^{(\log_3 7)^2}$, raise both sides of the given equation to the power of $\log_3 7$: $$ (a^{\log_3 7})^{\log_3 7} = 27^{\log_3 7} $$ $$ a^{(\log_3 7)^2} = 27^{\log_3 7} $$ Since $27 = 3^3$, we can rewrite the right side as: $$ a^{(\log_3 7)^2} = (3^3)^{\log_3 7} = 3^{3 \log_3 7} $$ Using the logarithm property $k \log_b x = \log_b x^k$: $$ 3^{3 \log_3 7} = 3^{\log_3 (7^3)} $$ Applying the property $b^{\log_b x} = x$: $$ 3^{\log_3 (7^3)} = 7^3 = 343 $$ Thus, $a^{(\log_3 7)^2} = 343$. **Step 2: Determine the value of $b^{(\log_7 11)^2}$** Given the equation $b^{\log_7 11} = 49$. To find $b^{(\log_7 11)^2}$, raise both sides of the given equation to the power of $\log_7 11$: $$ (b^{\log_7 11})^{\log_7 11} = 49^{\log_7 11} $$ $$ b^{(\log_7 11)^2} = 49^{\log_7 11} $$ Since $49 = 7^2$, we can rewrite the right side as: $$ b^{(\log_7 11)^2} = (7^2)^{\log_7 11} = 7^{2 \log_7 11} $$ Using the logarithm property $k \log_b x = \log_b x^k$: $$ 7^{2 \log_7 11} = 7^{\log_7 (11^2)} $$ Applying the property $b^{\log_b x} = x$: $$ 7^{\log_7 (11^2)} = 11^2 = 121 $$ Thus, $b^{(\log_7 11)^2} = 121$. **Step 3: Determine the value of $c^{(\log_{11} 25)^2}$** Given the equation $c^{\log_{11} 25} = 11$. To find $c^{(\log_{11} 25)^2}$, raise both sides of the given equation to the power of $\log_{11} 25$: $$ (c^{\log_{11} 25})^{\log_{11} 25} = 11^{\log_{11} 25} $$ $$ c^{(\log_{11} 25)^2} = 11^{\log_{11} 25} $$ Applying the property $b^{\log_b x} = x$: $$ 11^{\log_{11} 25} = 25 $$ Thus, $c^{(\log_{11} 25)^2} = 25$. **Step 4: Calculate the value of $S$** The expression for $S$ is given by: $$ S = a^{(\log_3 7)^2} + b^{(\log_7 11)^2} + c^{(\log_{11} 25)^2} $$ Substitute the values found in Steps 1, 2, and 3: $$ S = 343 + 121 + 25 $$ $$ S = 489 $$ **Step 5: Find the sum of the digits of $S$** The sum of the digits of $S = 489$ is: $$ 4 + 8 + 9 = 21 $$