a) $$ \frac{15^{n+2} \cdot 45^{1-n}}{3^{3-n}} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{15^{n+2} \cdot 45^{1-n}}{3^{3-n}}
$$

**Step 1: Express all numbers in terms of prime factors (3 and 5).**

- $$15 = 3 \times 5$$
- $$45 = 3^2 \times 5$$

**Step 2: Rewrite the expression using these prime factors.**

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

**Step 3: Apply the power rule $$(a \times b)^c = a^c \times b^c$$.**

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

**Step 4: Simplify the exponents for the base 3 and base 5 separately.**

- **For base 3:**
  $$
  3^{n+2} \times 3^{2-2n} \times 3^{n-3} = 3^{(n+2) + (2-2n) + (n-3)} = 3^{1}
  $$

- **For base 5:**
  $$
  5^{n+2} \times 5^{1-n} = 5^{(n+2) + (1-n)} = 5^{3}
  $$

**Step 5: Combine the simplified terms.**

$$
3^1 \times 5^3 = 3 \times 125 = 375
$$

**Final Simplified Form:**

$$
375
$$
The expression simplifies to 375.

a)