a) Step 1: Rewrite the terms in the numerator and denominator using exponent rules $a^{m+n} = a^m \cdot a^n$. $$ \frac{3^{n+1} + 9 \cdot 3^n}{6 \cdot 3^n + 3^{n+1}} = \frac{3^n \cdot 3^1 + 9 \cdot 3^n}{6 \cdot 3^n + 3^n \cdot 3^1} $$ Step 2: Factor out the common term $3^n$ from the numerator and the denominator. $$ \frac{3^n(3 + 9)}{3^n(6 + 3)} $$ Step 3: Simplify the expression by canceling out $3^n$ and performing the additions. $$ \frac{3 + 9}{6 + 3} = \frac{12}{9} $$ Step 4: Reduce the fraction to its simplest form. $$ \frac{12}{9} = \frac{4}{3} $$ The simplified expression is $\boxed{\frac{4}{3}}$. b) Step 1: Rewrite the terms using exponent rules $a^{m+n} = a^m \cdot a^n$ and $a^{xy} = (a^x)^y$. Note that $25^x = (5^2)^x = 5^{2x}$. $$ \frac{5^{2x+1} - 25^x}{3 \cdot 5^{2x+1} + 25^x} = \frac{5^{2x} \cdot 5^1 - 5^{2x}}{3 \cdot 5^{2x} \cdot 5^1 + 5^{2x}} $$ Step 2: Factor out the common term $5^{2x}$ from the numerator and the denominator. $$ \frac{5^{2x}(5 - 1)}{5^{2x}(3 \cdot 5 + 1)} $$ Step 3: Simplify the expression by canceling out $5^{2x}$ and performing the arithmetic. $$ \frac{5 - 1}{15 + 1} = \frac{4}{16} $$ Step 4: Reduce the fraction to its simplest form. $$ \frac{4}{16} = \frac{1}{4} $$ The simplified expression is $\boxed{\frac{1}{4}}$. That's 2 down. 3 left today — send the next one.