Question

If \[ \sqrt{12+4\sqrt5} + \sqrt{16-4\sqrt7} = \sqrt{x}+\sqrt{y}, \] then find the value of \[ \sqrt{35xy}. \]

• A. \(7\sqrt5\)
• B. \(14\sqrt5\)
• C. \(105\)
• D. \(70\)

Hint:

To simplify \(\sqrt{a\pm2\sqrt b}\), find two numbers whose sum is \(a\) and product is \(b\).

Answer 1

The Correct Option is D

Concept: Expressions of the form \[ \sqrt{a\pm2\sqrt b} \] can often be written as \[ \sqrt m \pm \sqrt n \] where \[ m+n=a,\qquad mn=b. \]

Step 1: Simplify \(\sqrt{12+4\sqrt5}\). Let \[ \sqrt{12+4\sqrt5} = \sqrt m+\sqrt n \] Then \[ m+n=12 \] and \[ 2\sqrt{mn}=4\sqrt5 \] \[ mn=20 \] Hence \[ m=10,\qquad n=2 \] Therefore, \[ \sqrt{12+4\sqrt5} = \sqrt{10}+\sqrt2 \]

Step 2: Simplify \(\sqrt{16-4\sqrt7}\). Let \[ \sqrt{16-4\sqrt7} = \sqrt m-\sqrt n \] Then \[ m+n=16 \] and \[ mn=28 \] Thus \[ m=14,\qquad n=2 \] Therefore, \[ \sqrt{16-4\sqrt7} = \sqrt{14}-\sqrt2 \]

Step 3: Add the two expressions. \[ (\sqrt{10}+\sqrt2)+(\sqrt{14}-\sqrt2) \] \[ =\sqrt{10}+\sqrt{14} \] Hence, \[ x=10,\qquad y=14 \]

Step 4: Compute \(\sqrt{35xy}\). \[ \sqrt{35xy} = \sqrt{35\times10\times14} \] \[ =\sqrt{4900} \] \[ =70 \] Therefore, \[ \boxed{70} \]