Question

Find the value of \( \sqrt{11 + \sqrt{72}} + \sqrt{11 - \sqrt{72}} \)

Hint:

For expressions of the form \(\sqrt{A + \sqrt{B}} \pm \sqrt{A - \sqrt{B}}\), squaring the entire expression is a very effective technique. The cross-product term \(2\sqrt{(A+\sqrt{B})(A-\sqrt{B})}\) simplifies nicely to \(2\sqrt{A^2 - B}\).

Answer 1

Step 1: Understanding the Concept:
This problem involves simplifying a sum of nested square roots. A common and efficient method is to let the entire expression equal a variable, say \(X\), and then square both sides. This often simplifies the expression by eliminating the outer square roots and using the difference of squares identity.
Step 2: Detailed Explanation:
Let \(X = \sqrt{11 + \sqrt{72}} + \sqrt{11 - \sqrt{72}}\).
Since \(X\) is a sum of positive square roots, \(X\) must be positive.
Now, square both sides of the equation: \[ X^2 = (\sqrt{11 + \sqrt{72}} + \sqrt{11 - \sqrt{72}})^2 \] Using the identity \((a+b)^2 = a^2 + b^2 + 2ab\), where \(a = \sqrt{11 + \sqrt{72}}\) and \(b = \sqrt{11 - \sqrt{72}}\): \[ X^2 = (\sqrt{11 + \sqrt{72}})^2 + (\sqrt{11 - \sqrt{72}})^2 + 2(\sqrt{11 + \sqrt{72}})(\sqrt{11 - \sqrt{72}}) \] \[ X^2 = (11 + \sqrt{72}) + (11 - \sqrt{72}) + 2\sqrt{(11 + \sqrt{72})(11 - \sqrt{72})} \] The \(\sqrt{72}\) terms cancel out. For the product under the square root, we use the difference of squares identity \((a+b)(a-b) = a^2 - b^2\). \[ X^2 = 22 + 2\sqrt{11^2 - (\sqrt{72})^2} \] \[ X^2 = 22 + 2\sqrt{121 - 72} \] \[ X^2 = 22 + 2\sqrt{49} \] \[ X^2 = 22 + 2(7) \] \[ X^2 = 22 + 14 \] \[ X^2 = 36 \] Step 3: Final Answer:
Taking the square root of both sides: \[ X = \sqrt{36} \] Since we established that \(X\) must be positive, we take the positive root. \[ X = 6 \]