Square root of $(7 - 4\sqrt{3})$ is equal to :

Q: Square root of $(7 - 4\sqrt{3})$ is equal to :

To determine the square root of $7 - 4\sqrt{3}$, we need to find a number that, when squared, equals $7 - 4\sqrt{3}$. The correct answer is given as Option 2: $\pm (2 - \sqrt{3})$. Let's verify this and explain why the other options are incorrect. ### Verification of Option 2: $\pm (2 - \sqrt{3})$ 1. **Calculate $(2 - \sqrt{3})^2$:** \[ (2 - \sqrt{3})^2 = (2)^2 - 2 \times 2 \times \sqrt{3} + (\sqrt{3})^2 \] \[ = 4 - 4\sqrt{3} + 3 \] \[ = 7 - 4\sqrt{3} \] This matches the original expression $7 - 4\sqrt{3}$, confirming that $(2 - \sqrt{3})$ is indeed a square root of $7 - 4\sqrt{3}$. Therefore, Option 2 is correct. ### Explanation of Incorrectness of Other Options: 1. **Option 1: $\pm (2 + \sqrt{3})$** - Calculate $(2 + \sqrt{3})^2$: \[ (2 + \sqrt{3})^2 = (2)^2 + 2 \times 2 \times \sqrt{3} + (\sqrt{3})^2 \] \[ = 4 + 4\sqrt{3} + 3 \] \[ = 7 + 4\sqrt{3} \] - This result is $7 + 4\sqrt{3}$, not $7 - 4\sqrt{3}$. Therefore, Option 1 is incorrect. 2. **Option 3: $\pm (\sqrt{2} - 3)$** - Calculate $(\sqrt{2} - 3)^2$: \[ (\sqrt{2} - 3)^2 = (\sqrt{2})^2 - 2 \times \sqrt{2} \times 3 + (3)^2 \] \[ = 2 - 6\sqrt{2} + 9 \] \[ = 11 - 6\sqrt{2} \] - This result is $11 - 6\sqrt{2}$, not $7 - 4\sqrt{3}$. Therefore, Option 3 is incorrect. 3. **Option 4: $\pm (\sqrt{2} + 3)$** - Calculate $(\sqrt{2} + 3)^2$: \[ (\sqrt{2} + 3)^2 = (\sqrt{2})^2 + 2 \times \sqrt{2} \times 3 + (3)^2 \] \[ = 2 + 6\sqrt{2} + 9 \] \[ = 11 + 6\sqrt{2} \] - This result is $11 + 6\sqrt{2}$, not $7 - 4\sqrt{3}$. Therefore, Option 4 is incorrect. In conclusion, the only option that correctly squares to $7 - 4\sqrt{3}$ is Option 2: $\pm (2 - \sqrt{3})$.

Question

Accepted Answer

To determine the square root of $7 - 4\sqrt{3}$, we need to find a number that, when squared, equals $7 - 4\sqrt{3}$. The correct answer is given as Option 2: $\pm (2 - \sqrt{3})$. Let's verify this and explain why the other options are incorrect.

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

1. **Calculate $(2 - \sqrt{3})^2$:**

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

$$
   = 4 - 4\sqrt{3} + 3
   $$

$$
   = 7 - 4\sqrt{3}
   $$

This matches the original expression $7 - 4\sqrt{3}$, confirming that $(2 - \sqrt{3})$ is indeed a square root of $7 - 4\sqrt{3}$. Therefore, Option 2 is correct.

### Explanation of Incorrectness of Other Options:

1. **Option 1: $\pm (2 + \sqrt{3})$**

- Calculate $(2 + \sqrt{3})^2$:

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

$$
     = 4 + 4\sqrt{3} + 3
     $$

$$
     = 7 + 4\sqrt{3}
     $$

- This result is $7 + 4\sqrt{3}$, not $7 - 4\sqrt{3}$. Therefore, Option 1 is incorrect.

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

- Calculate $(\sqrt{2} - 3)^2$:

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

$$
     = 2 - 6\sqrt{2} + 9
     $$

$$
     = 11 - 6\sqrt{2}
     $$

- This result is $11 - 6\sqrt{2}$, not $7 - 4\sqrt{3}$. Therefore, Option 3 is incorrect.

3. **Option 4: $\pm (\sqrt{2} + 3)$**

- Calculate $(\sqrt{2} + 3)^2$:

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

$$
     = 2 + 6\sqrt{2} + 9
     $$

$$
     = 11 + 6\sqrt{2}
     $$

- This result is $11 + 6\sqrt{2}$, not $7 - 4\sqrt{3}$. Therefore, Option 4 is incorrect.

In conclusion, the only option that correctly squares to $7 - 4\sqrt{3}$ is Option 2: $\pm (2 - \sqrt{3})$.