c) $$ \frac{5^{2 n+1} \cdot 3^{2 n-3}}{15^{2 n}} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{5^{2n+1} \cdot 3^{2n-3}}{15^{2n}}
$$

**Step 1: Break Down the Denominator**

First, express $$15^{2n}$$ in terms of its prime factors:

$$
15^{2n} = (3 \cdot 5)^{2n} = 3^{2n} \cdot 5^{2n}
$$

**Step 2: Rewrite the Original Expression**

Substitute the expression for $$15^{2n}$$ back into the original fraction:

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

**Step 3: Simplify the Fraction**

Simplify by subtracting the exponents of like bases:

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

**Step 4: Simplify Further**

Recall that $$3^{-3} = \frac{1}{3^3} = \frac{1}{27}$$. Therefore:

$$
5^{1} \cdot 3^{-3} = 5 \cdot \frac{1}{27} = \frac{5}{27}
$$

**Final Answer:**

$$
\frac{5}{27}
$$
The simplified form of the expression is $$ \frac{5}{27} $$.

c)