$ ext{ If } \frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}} = a - b\sqrt{77}, ext{ then } (a+b) ext{ is equal to :}$

Q: $\text{ If } \frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}} = a - b\sqrt{77}, \text{ then } (a+b) \text{ is equal to :}$

To solve the problem, we need to simplify the expression $\frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}}$ and express it in the form $a - b\sqrt{77}$. Then, we will find the sum $a + b$. ### Step-by-step Solution: 1. **Rationalize the Denominator:** To simplify the expression, we multiply the numerator and the denominator by the conjugate of the denominator, which is $\sqrt{11} - \sqrt{7}$. \[ \frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}} \times \frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}-\sqrt{7}} = \frac{(\sqrt{11}-\sqrt{7})^2}{(\sqrt{11})^2 - (\sqrt{7})^2} \] 2. **Simplify the Denominator:** \[ (\sqrt{11})^2 - (\sqrt{7})^2 = 11 - 7 = 4 \] 3. **Expand the Numerator:** \[ (\sqrt{11}-\sqrt{7})^2 = (\sqrt{11})^2 - 2\sqrt{11}\sqrt{7} + (\sqrt{7})^2 = 11 - 2\sqrt{77} + 7 = 18 - 2\sqrt{77} \] 4. **Combine the Results:** \[ \frac{18 - 2\sqrt{77}}{4} = \frac{18}{4} - \frac{2\sqrt{77}}{4} = \frac{9}{2} - \frac{\sqrt{77}}{2} \] This expression is in the form $a - b\sqrt{77}$, where $a = \frac{9}{2}$ and $b = \frac{1}{2}$. 5. **Calculate $a + b$:** \[ a + b = \frac{9}{2} + \frac{1}{2} = \frac{10}{2} = 5 \] Thus, the value of $a + b$ is 5, which corresponds to Option 1. ### Explanation of Incorrect Options: - **Option 2 (7):** This option is incorrect because the calculation of $a + b$ results in 5, not 7. The arithmetic operations and simplifications do not lead to 7. - **Option 3 (11):** This option is incorrect because the sum $a + b$ is derived from the simplified expression $\frac{9}{2} - \frac{\sqrt{77}}{2}$, which clearly adds up to 5, not 11. - **Option 4 (18):** This option is incorrect because 18 is the result of the initial expansion of the numerator before division by 4. The final sum $a + b$ after simplification is 5, not 18. In conclusion, the correct answer is Option 1, as the sum $a + b$ is indeed 5.

Question

$	ext{ If } \frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}} = a - b\sqrt{77}, 	ext{ then } (a+b) 	ext{ is equal to :}$

Accepted Answer

To solve the problem, we need to simplify the expression $\frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}}$ and express it in the form $a - b\sqrt{77}$. Then, we will find the sum $a + b$.

### Step-by-step Solution:

1. **Rationalize the Denominator:**
   To simplify the expression, we multiply the numerator and the denominator by the conjugate of the denominator, which is $\sqrt{11} - \sqrt{7}$.

$$
   \frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}} \times \frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}-\sqrt{7}} = \frac{(\sqrt{11}-\sqrt{7})^2}{(\sqrt{11})^2 - (\sqrt{7})^2}
   $$

2. **Simplify the Denominator:**
   $$
   (\sqrt{11})^2 - (\sqrt{7})^2 = 11 - 7 = 4
   $$

3. **Expand the Numerator:**
   $$
   (\sqrt{11}-\sqrt{7})^2 = (\sqrt{11})^2 - 2\sqrt{11}\sqrt{7} + (\sqrt{7})^2 = 11 - 2\sqrt{77} + 7 = 18 - 2\sqrt{77}
   $$

4. **Combine the Results:**
   $$
   \frac{18 - 2\sqrt{77}}{4} = \frac{18}{4} - \frac{2\sqrt{77}}{4} = \frac{9}{2} - \frac{\sqrt{77}}{2}
   $$

This expression is in the form $a - b\sqrt{77}$, where $a = \frac{9}{2}$ and $b = \frac{1}{2}$.

5. **Calculate $a + b$:**
   $$
   a + b = \frac{9}{2} + \frac{1}{2} = \frac{10}{2} = 5
   $$

Thus, the value of $a + b$ is 5, which corresponds to Option 1.

### Explanation of Incorrect Options:

- **Option 2 (7):** This option is incorrect because the calculation of $a + b$ results in 5, not 7. The arithmetic operations and simplifications do not lead to 7.

- **Option 3 (11):** This option is incorrect because the sum $a + b$ is derived from the simplified expression $\frac{9}{2} - \frac{\sqrt{77}}{2}$, which clearly adds up to 5, not 11.

- **Option 4 (18):** This option is incorrect because 18 is the result of the initial expansion of the numerator before division by 4. The final sum $a + b$ after simplification is 5, not 18.

In conclusion, the correct answer is Option 1, as the sum $a + b$ is indeed 5.