If $\sqrt{3^n}$ = 729, then the value of n is :

Q: If $\sqrt{3^n}$ = 729, then the value of n is :

To solve the problem, we need to find the value of $ n $ such that $\sqrt{3^n} = 729$. First, let's rewrite the equation using the property of square roots: \[ \sqrt{3^n} = (3^n)^{1/2} = 3^{n/2} \] We are given that: \[ 3^{n/2} = 729 \] Next, we express 729 as a power of 3. We know: \[ 729 = 3^6 \] Thus, the equation becomes: \[ 3^{n/2} = 3^6 \] Since the bases are the same, we can equate the exponents: \[ \frac{n}{2} = 6 \] Solving for $ n $, we multiply both sides by 2: \[ n = 12 \] Therefore, the value of $ n $ is 12, which corresponds to Option 4. **Explanation of Incorrect Options:** 1. **Option 1 (6):** If $ n = 6 $, then $\sqrt{3^6} = 3^{6/2} = 3^3 = 27$, which is not equal to 729. Therefore, Option 1 is incorrect. 2. **Option 2 (8):** If $ n = 8 $, then $\sqrt{3^8} = 3^{8/2} = 3^4 = 81$, which is not equal to 729. Therefore, Option 2 is incorrect. 3. **Option 3 (10):** If $ n = 10 $, then $\sqrt{3^{10}} = 3^{10/2} = 3^5 = 243$, which is not equal to 729. Therefore, Option 3 is incorrect. In conclusion, the only value of $ n $ that satisfies the equation $\sqrt{3^n} = 729$ is 12, making Option 4 the correct answer.

Question

Accepted Answer

To solve the problem, we need to find the value of $ n $ such that $\sqrt{3^n} = 729$.

First, let's rewrite the equation using the property of square roots:

$$
\sqrt{3^n} = (3^n)^{1/2} = 3^{n/2}
$$

We are given that:

$$
3^{n/2} = 729
$$

Next, we express 729 as a power of 3. We know:

$$
729 = 3^6
$$

Thus, the equation becomes:

$$
3^{n/2} = 3^6
$$

Since the bases are the same, we can equate the exponents:

$$
\frac{n}{2} = 6
$$

Solving for $ n $, we multiply both sides by 2:

$$
n = 12
$$

Therefore, the value of $ n $ is 12, which corresponds to Option 4.

**Explanation of Incorrect Options:**

1. **Option 1 (6):** If $ n = 6 $, then $\sqrt{3^6} = 3^{6/2} = 3^3 = 27$, which is not equal to 729. Therefore, Option 1 is incorrect.

2. **Option 2 (8):** If $ n = 8 $, then $\sqrt{3^8} = 3^{8/2} = 3^4 = 81$, which is not equal to 729. Therefore, Option 2 is incorrect.

3. **Option 3 (10):** If $ n = 10 $, then $\sqrt{3^{10}} = 3^{10/2} = 3^5 = 243$, which is not equal to 729. Therefore, Option 3 is incorrect.

In conclusion, the only value of $ n $ that satisfies the equation $\sqrt{3^n} = 729$ is 12, making Option 4 the correct answer.