Group terms with the same base.

Here are the step-by-step solutions for each expression:

i)

Step 1: Group terms with the same base.

$$\frac{5^{2n+1}}{5^{2n}} \times \frac{3^{2n-3}}{3^{2n}}$$

Step 2: Apply the quotient rule for exponents, $\frac{a^m}{a^n} = a^{m-n}$.

$$5^{(2n+1) - 2n} \times 3^{(2n-3) - 2n}$$

Step 3: Simplify the exponents.

$$5^{1} \times 3^{-3}$$

Step 4: Evaluate the powers.

$$5 \times \frac{1}{3^3} = 5 \times \frac{1}{27} = \frac{5}{27}$$

The simplified expression is $\boxed{\frac{5}{27}}$.

j)

Step 1: Combine terms in the numerator with base 5 using $a^m \times a^n = a^{m+n}$. Note that $5 = 5^1$.

$$\frac{5^{2n+1 + (2n-2) + 1}}{25^{2n-1}} = \frac{5^{4n}}{25^{2n-1}}$$

Step 2: Express the denominator with base 5, since $25 = 5^2$.

$$\frac{5^{4n}}{(5^2)^{2n-1}}$$

Step 3: Apply the power of a power rule, $(a^m)^n = a^{mn}$.

$$\frac{5^{4n}}{5^{2(2n-1)}} = \frac{5^{4n}}{5^{4n-2}}$$

Step 4: Apply the quotient rule for exponents, $\frac{a^m}{a^n} = a^{m-n}$.

$$5^{4n - (4n-2)} = 5^{4n - 4n + 2} = 5^2$$

Step 5: Evaluate the power.

$$25$$

The simplified expression is $\boxed{25}$.

k)

Step 1: Express all bases as products of their prime factors: $12 = 2^2 \times 3$, $9 = 3^2$, $36 = 2^2 \times 3^2$, $8 = 2^3$.

$$\frac{(2^2 \times 3)^{n+1} \times (3^2)^{2n-1}}{(2^2 \times 3^2)^n \times (2^3)^{1-n}}$$

Step 2: Apply the power rules $(ab)^m = a^m b^m$ and $(a^m)^n = a^{mn}$.

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