If $3^{(x+8)} = 27^{(2x+1)}$, then the value of x is :

Q: If $3^{(x+8)} = 27^{(2x+1)}$, then the value of x is :

To solve the equation $3^{(x+8)} = 27^{(2x+1)}$ and determine the value of $x$, we need to express both sides of the equation with the same base. First, note that 27 can be expressed as a power of 3: \[ 27 = 3^3. \] Therefore, we can rewrite the equation as: \[ 3^{(x+8)} = (3^3)^{(2x+1)}. \] Using the property of exponents $(a^m)^n = a^{m \cdot n}$, we can simplify the right side: \[ (3^3)^{(2x+1)} = 3^{3(2x+1)}. \] This simplifies to: \[ 3^{(x+8)} = 3^{(6x+3)}. \] Since the bases are the same, we can equate the exponents: \[ x + 8 = 6x + 3. \] Now, solve for $x$: 1. Subtract $x$ from both sides: \[ 8 = 5x + 3. \] 2. Subtract 3 from both sides: \[ 5 = 5x. \] 3. Divide both sides by 5: \[ x = 1. \] Thus, the value of $x$ is 1, which corresponds to Option 2. **Explanation for Incorrect Options:** - **Option 1 (x = 0):** If $x = 0$, substituting into the equation gives $3^{(0+8)} = 27^{(2 \cdot 0 + 1)}$, which simplifies to $3^8 = 27^1$. This is incorrect because $3^8$ is not equal to $27$. - **Option 3 (x = 2):** If $x = 2$, substituting into the equation gives $3^{(2+8)} = 27^{(2 \cdot 2 + 1)}$, which simplifies to $3^{10} = 27^5$. This is incorrect because $3^{10}$ is not equal to $27^5$. - **Option 4 (x = 3):** If $x = 3$, substituting into the equation gives $3^{(3+8)} = 27^{(2 \cdot 3 + 1)}$, which simplifies to $3^{11} = 27^7$. This is incorrect because $3^{11}$ is not equal to $27^7$. Therefore, the only correct option is Option 2, where $x = 1$.

Question

Accepted Answer

To solve the equation $3^{(x+8)} = 27^{(2x+1)}$ and determine the value of $x$, we need to express both sides of the equation with the same base.

First, note that 27 can be expressed as a power of 3: 
$$ 27 = 3^3. $$

Therefore, we can rewrite the equation as:
$$ 3^{(x+8)} = (3^3)^{(2x+1)}. $$

Using the property of exponents $(a^m)^n = a^{m \cdot n}$, we can simplify the right side:
$$ (3^3)^{(2x+1)} = 3^{3(2x+1)}. $$

This simplifies to:
$$ 3^{(x+8)} = 3^{(6x+3)}. $$

Since the bases are the same, we can equate the exponents:
$$ x + 8 = 6x + 3. $$

Now, solve for $x$:
1. Subtract $x$ from both sides:
   $$ 8 = 5x + 3. $$

2. Subtract 3 from both sides:
   $$ 5 = 5x. $$

3. Divide both sides by 5:
   $$ x = 1. $$

Thus, the value of $x$ is 1, which corresponds to Option 2.

**Explanation for Incorrect Options:**

- **Option 1 (x = 0):** If $x = 0$, substituting into the equation gives $3^{(0+8)} = 27^{(2 \cdot 0 + 1)}$, which simplifies to $3^8 = 27^1$. This is incorrect because $3^8$ is not equal to $27$.

- **Option 3 (x = 2):** If $x = 2$, substituting into the equation gives $3^{(2+8)} = 27^{(2 \cdot 2 + 1)}$, which simplifies to $3^{10} = 27^5$. This is incorrect because $3^{10}$ is not equal to $27^5$.

- **Option 4 (x = 3):** If $x = 3$, substituting into the equation gives $3^{(3+8)} = 27^{(2 \cdot 3 + 1)}$, which simplifies to $3^{11} = 27^7$. This is incorrect because $3^{11}$ is not equal to $27^7$.

Therefore, the only correct option is Option 2, where $x = 1$.