What is the value of `sqrt(18-2sqrt(77))`?

Options

A. `sqrt(11)+sqrt(7)`
B. `sqrt(12)-sqrt(5)`
C. `sqrt(11)-sqrt(7)`
D. `sqrt(13)-sqrt(4)`

Answer

To find the value of \( \sqrt{18 - 2\sqrt{77}} \), we can express the expression under the square root in a different form.

### Step-by-Step Solution:

1. **Identify the structure**: We notice that the expression \( 18 - 2\sqrt{77} \) resembles the form of a squared binomial, \( (a - b)^2 = a^2 - 2ab \).

2. **Set up the equation**: We can assume that \( \sqrt{18 - 2\sqrt{77}} \) can be expressed as \( \sqrt{(a - b)^2} = a - b \). Thus, we need to find \( a \) and \( b \) such that:
\[ a^2 + b^2 = 18 \quad \text{and} \quad 2ab = 2\sqrt{77} \]

3. **Solve for \( ab \)**: From the second equation \( 2ab = 2\sqrt{77} \), we can simplify this to:
\[ ab = \sqrt{77} \]

4. **Express \( b \) in terms of \( a \)**: We can express \( b \) as:
\[ b = \frac{\sqrt{77}}{a} \]

5. **Substitute \( b \) into the first equation**: Substitute \( b \) into the first equation:
\[ a^2 + \left(\frac{\sqrt{77}}{a}\right)^2 = 18 \]
This simplifies to:
\[ a^2 + \frac{77}{a^2} = 18 \]

6. **Multiply through by \( a^2 \)**: To eliminate the fraction, multiply through by \( a^2 \):
\[ a^4 - 18a^2 + 77 = 0 \]

7. **Let \( x = a^2 \)**: This gives us a quadratic equation:
\[ x^2 - 18x + 77 = 0 \]

8. **Use the quadratic formula**: The solutions for \( x \) are given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{18 \pm \sqrt{(-18)^2 - 4 \cdot 1 \cdot 77}}{2 \cdot 1} \]
\[ = \frac{18 \pm \sqrt{324 - 308}}{2} = \frac{18 \pm \sqrt{16}}{2} = \frac{18 \pm 4}{2} \]

9. **Calculate the roots**: This gives us:
\[ x = \frac{22}{2} = 11 \quad \text{and} \quad x = \frac{14}{2} = 7 \]

10. **Find \( a \) and \( b \)**: Thus, \( a^2 = 11 \) and \( b^2 = 7 \) (or vice versa). Therefore:
\[ a = \sqrt{11}, \quad b = \sqrt{7} \]

11. **Final result**: Now substituting back, we have:
\[ \sqrt{18 - 2\sqrt{77}} = \sqrt{11} - \sqrt{7} \]

### Conclusion:
The value of \( \sqrt{18 - 2\sqrt{77}} \) is \( \sqrt{11} - \sqrt{7} \).