Find the value of X, if $5^{x-3} \cdot 3^{2x-8}$ = 225 :

Q: Find the value of X, if $5^{x-3} \cdot 3^{2x-8}$ = 225 :

To solve the equation $5^{x-3} \cdot 3^{2x-8} = 225$ and verify why Option 2 (x = 5) is correct, let's break down the problem step by step. ### Step-by-Step Solution: 1. **Express 225 in terms of its prime factors:** \[ 225 = 15^2 = (3 \times 5)^2 = 3^2 \times 5^2 \] So, $225 = 3^2 \times 5^2$. 2. **Set up the equation with the prime factorization:** \[ 5^{x-3} \cdot 3^{2x-8} = 3^2 \times 5^2 \] 3. **Equate the exponents of the same bases:** - For the base 5: \[ x - 3 = 2 \] Solving for $x$, we get: \[ x = 5 \] - For the base 3: \[ 2x - 8 = 2 \] Solving for $x$, we get: \[ 2x = 10 \implies x = 5 \] Both calculations confirm that $x = 5$. ### Explanation of Why Other Options are Incorrect: - **Option 1 (x = 3):** - If $x = 3$, then: \[ 5^{3-3} \cdot 3^{2 \times 3 - 8} = 5^0 \cdot 3^{-2} = 1 \cdot \frac{1}{9} = \frac{1}{9} \] - This does not equal 225, so Option 1 is incorrect. - **Option 3 (x = 6):** - If $x = 6$, then: \[ 5^{6-3} \cdot 3^{2 \times 6 - 8} = 5^3 \cdot 3^4 = 125 \cdot 81 \] - This equals 10125, which is not 225, so Option 3 is incorrect. - **Option 4 (x = 7):** - If $x = 7$, then: \[ 5^{7-3} \cdot 3^{2 \times 7 - 8} = 5^4 \cdot 3^6 = 625 \cdot 729 \] - This equals 456,625, which is not 225, so Option 4 is incorrect. ### Conclusion: The calculations clearly show that when $x = 5$, the equation $5^{x-3} \cdot 3^{2x-8} = 225$ holds true. Therefore, Option 2 is indeed the correct answer.

Question

Accepted Answer

To solve the equation $5^{x-3} \cdot 3^{2x-8} = 225$ and verify why Option 2 (x = 5) is correct, let's break down the problem step by step.

### Step-by-Step Solution:

1. **Express 225 in terms of its prime factors:**
   $$
   225 = 15^2 = (3 \times 5)^2 = 3^2 \times 5^2
   $$
   So, $225 = 3^2 \times 5^2$.

2. **Set up the equation with the prime factorization:**
   $$
   5^{x-3} \cdot 3^{2x-8} = 3^2 \times 5^2
   $$

3. **Equate the exponents of the same bases:**
   - For the base 5:
     $$
     x - 3 = 2
     $$
     Solving for $x$, we get:
     $$
     x = 5
     $$

- For the base 3:
     $$
     2x - 8 = 2
     $$
     Solving for $x$, we get:
     $$
     2x = 10 \implies x = 5
     $$

Both calculations confirm that $x = 5$.

### Explanation of Why Other Options are Incorrect:

- **Option 1 (x = 3):**
  - If $x = 3$, then:
    $$
    5^{3-3} \cdot 3^{2 \times 3 - 8} = 5^0 \cdot 3^{-2} = 1 \cdot \frac{1}{9} = \frac{1}{9}
    $$
  - This does not equal 225, so Option 1 is incorrect.

- **Option 3 (x = 6):**
  - If $x = 6$, then:
    $$
    5^{6-3} \cdot 3^{2 \times 6 - 8} = 5^3 \cdot 3^4 = 125 \cdot 81
    $$
  - This equals 10125, which is not 225, so Option 3 is incorrect.

- **Option 4 (x = 7):**
  - If $x = 7$, then:
    $$
    5^{7-3} \cdot 3^{2 \times 7 - 8} = 5^4 \cdot 3^6 = 625 \cdot 729
    $$
  - This equals 456,625, which is not 225, so Option 4 is incorrect.

### Conclusion:

The calculations clearly show that when $x = 5$, the equation $5^{x-3} \cdot 3^{2x-8} = 225$ holds true. Therefore, Option 2 is indeed the correct answer.