If 27^{x - \frac{2}{3}} = 81, then value of 2^{3x+1} is :

Q: If 27^{x - \frac{2}{3}} = 81, then value of 2^{3x+1} is :

To solve the problem, we need to find the value of $2^{3x+1}$ given that $27^{x - \frac{2}{3}} = 81$. **Step 1: Solve for $x$ in the equation $27^{x - \frac{2}{3}} = 81$.** First, express 27 and 81 as powers of 3: - $27 = 3^3$ - $81 = 3^4$ Substitute these into the equation: \[ (3^3)^{x - \frac{2}{3}} = 3^4 \] Simplify the left side using the power of a power property $(a^m)^n = a^{m \cdot n}$: \[ 3^{3(x - \frac{2}{3})} = 3^4 \] Since the bases are the same, we can equate the exponents: \[ 3(x - \frac{2}{3}) = 4 \] Distribute the 3: \[ 3x - 2 = 4 \] Solve for $x$: \[ 3x = 6 \] \[ x = 2 \] **Step 2: Find the value of $2^{3x+1}$.** Substitute $x = 2$ into $2^{3x+1}$: \[ 2^{3(2)+1} = 2^{6+1} = 2^7 \] Calculate $2^7$: \[ 2^7 = 128 \] Thus, the value of $2^{3x+1}$ is 128, which corresponds to Option 2. **Explanation of Incorrect Options:** - **Option 1 (64):** This would imply $2^6$, which does not match the calculation for $2^{3x+1}$ when $x = 2$. - **Option 3 (256):** This would imply $2^8$, which is incorrect as $2^{3x+1} = 2^7$. - **Option 4 (512):** This would imply $2^9$, which is also incorrect for the same reason. Therefore, the correct answer is Option 2, which is 128.

Question

Accepted Answer

To solve the problem, we need to find the value of $2^{3x+1}$ given that $27^{x - \frac{2}{3}} = 81$.

**Step 1: Solve for $x$ in the equation $27^{x - \frac{2}{3}} = 81$.**

First, express 27 and 81 as powers of 3:
- $27 = 3^3$
- $81 = 3^4$

Substitute these into the equation:
$$
(3^3)^{x - \frac{2}{3}} = 3^4
$$

Simplify the left side using the power of a power property $(a^m)^n = a^{m \cdot n}$:
$$
3^{3(x - \frac{2}{3})} = 3^4
$$

Since the bases are the same, we can equate the exponents:
$$
3(x - \frac{2}{3}) = 4
$$

Distribute the 3:
$$
3x - 2 = 4
$$

Solve for $x$:
$$
3x = 6
$$
$$
x = 2
$$

**Step 2: Find the value of $2^{3x+1}$.**

Substitute $x = 2$ into $2^{3x+1}$:
$$
2^{3(2)+1} = 2^{6+1} = 2^7
$$

Calculate $2^7$:
$$
2^7 = 128
$$

Thus, the value of $2^{3x+1}$ is 128, which corresponds to Option 2.

**Explanation of Incorrect Options:**

- **Option 1 (64):** This would imply $2^6$, which does not match the calculation for $2^{3x+1}$ when $x = 2$.
- **Option 3 (256):** This would imply $2^8$, which is incorrect as $2^{3x+1} = 2^7$.
- **Option 4 (512):** This would imply $2^9$, which is also incorrect for the same reason.

Therefore, the correct answer is Option 2, which is 128.