$ ext{Square root of } (14 - 6\sqrt{5}) ext{ is :}$

Q: $\text{Square root of } (14 - 6\sqrt{5}) \text{ is :}$

To determine the square root of $14 - 6\sqrt{5}$, we need to find a number that, when squared, equals $14 - 6\sqrt{5}$. The correct answer is given as Option 2: $\pm (3 - \sqrt{5})$. Let's verify this and explain why this is the correct choice. ### Verification of Option 2: $\pm (3 - \sqrt{5})$ 1. **Calculate the square of $(3 - \sqrt{5})$:** \[ (3 - \sqrt{5})^2 = 3^2 - 2 \times 3 \times \sqrt{5} + (\sqrt{5})^2 \] \[ = 9 - 6\sqrt{5} + 5 \] \[ = 14 - 6\sqrt{5} \] The calculation shows that $(3 - \sqrt{5})^2$ indeed equals $14 - 6\sqrt{5}$. Therefore, $\pm (3 - \sqrt{5})$ is the correct square root of $14 - 6\sqrt{5}$. ### Explanation of Incorrect Options 1. **Option 1: $\pm (3 + \sqrt{5})$** - **Calculate the square of $(3 + \sqrt{5})$:** \[ (3 + \sqrt{5})^2 = 3^2 + 2 \times 3 \times \sqrt{5} + (\sqrt{5})^2 \] \[ = 9 + 6\sqrt{5} + 5 \] \[ = 14 + 6\sqrt{5} \] - This result, $14 + 6\sqrt{5}$, is not equal to $14 - 6\sqrt{5}$, so Option 1 is incorrect. 2. **Option 3: $\pm (\sqrt{3} - 3)$** - **Calculate the square of $(\sqrt{3} - 3)$:** \[ (\sqrt{3} - 3)^2 = (\sqrt{3})^2 - 2 \times \sqrt{3} \times 3 + 3^2 \] \[ = 3 - 6\sqrt{3} + 9 \] \[ = 12 - 6\sqrt{3} \] - This result, $12 - 6\sqrt{3}$, is not equal to $14 - 6\sqrt{5}$, so Option 3 is incorrect. 3. **Option 4: $\pm (\sqrt{3} + 3)$** - **Calculate the square of $(\sqrt{3} + 3)$:** \[ (\sqrt{3} + 3)^2 = (\sqrt{3})^2 + 2 \times \sqrt{3} \times 3 + 3^2 \] \[ = 3 + 6\sqrt{3} + 9 \] \[ = 12 + 6\sqrt{3} \] - This result, $12 + 6\sqrt{3}$, is not equal to $14 - 6\sqrt{5}$, so Option 4 is incorrect. In conclusion, the only option that satisfies the condition of being the square root of $14 - 6\sqrt{5}$ is Option 2: $\pm (3 - \sqrt{5})$.

Question

$	ext{Square root of } (14 - 6\sqrt{5}) 	ext{ is :}$

Accepted Answer

To determine the square root of $14 - 6\sqrt{5}$, we need to find a number that, when squared, equals $14 - 6\sqrt{5}$. The correct answer is given as Option 2: $\pm (3 - \sqrt{5})$. Let's verify this and explain why this is the correct choice.

### Verification of Option 2: $\pm (3 - \sqrt{5})$

1. **Calculate the square of $(3 - \sqrt{5})$:**

$$
   (3 - \sqrt{5})^2 = 3^2 - 2 \times 3 \times \sqrt{5} + (\sqrt{5})^2
   $$

$$
   = 9 - 6\sqrt{5} + 5
   $$

$$
   = 14 - 6\sqrt{5}
   $$

The calculation shows that $(3 - \sqrt{5})^2$ indeed equals $14 - 6\sqrt{5}$. Therefore, $\pm (3 - \sqrt{5})$ is the correct square root of $14 - 6\sqrt{5}$.

### Explanation of Incorrect Options

1. **Option 1: $\pm (3 + \sqrt{5})$**

- **Calculate the square of $(3 + \sqrt{5})$:**

$$
     (3 + \sqrt{5})^2 = 3^2 + 2 \times 3 \times \sqrt{5} + (\sqrt{5})^2
     $$

$$
     = 9 + 6\sqrt{5} + 5
     $$

$$
     = 14 + 6\sqrt{5}
     $$

- This result, $14 + 6\sqrt{5}$, is not equal to $14 - 6\sqrt{5}$, so Option 1 is incorrect.

2. **Option 3: $\pm (\sqrt{3} - 3)$**

- **Calculate the square of $(\sqrt{3} - 3)$:**

$$
     (\sqrt{3} - 3)^2 = (\sqrt{3})^2 - 2 \times \sqrt{3} \times 3 + 3^2
     $$

$$
     = 3 - 6\sqrt{3} + 9
     $$

$$
     = 12 - 6\sqrt{3}
     $$

- This result, $12 - 6\sqrt{3}$, is not equal to $14 - 6\sqrt{5}$, so Option 3 is incorrect.

3. **Option 4: $\pm (\sqrt{3} + 3)$**

- **Calculate the square of $(\sqrt{3} + 3)$:**

$$
     (\sqrt{3} + 3)^2 = (\sqrt{3})^2 + 2 \times \sqrt{3} \times 3 + 3^2
     $$

$$
     = 3 + 6\sqrt{3} + 9
     $$

$$
     = 12 + 6\sqrt{3}
     $$

- This result, $12 + 6\sqrt{3}$, is not equal to $14 - 6\sqrt{5}$, so Option 4 is incorrect.

In conclusion, the only option that satisfies the condition of being the square root of $14 - 6\sqrt{5}$ is Option 2: $\pm (3 - \sqrt{5})$.