Question

If \(a*b=\sqrt{ab}+\frac{1}{\sqrt{ab}}\) and \(a\odot b=\sqrt{ab}-\frac{1}{\sqrt{ab}}\), then find the value of \[ \frac{(4*5)+(4\odot5)} {(4*5)-(4\odot5)} \]

• A. \(20\)
• B. \(25\)
• C. \(\sqrt{\frac{21}{14}}\)
• D. \(\sqrt{\frac{18}{13}}\)

Hint:

When expressions appear as \((a+b)\pm(a-b)\), use cancellation immediately to simplify the calculation.

Answer 1

The Correct Option is A

Concept: Whenever two custom operations contain similar expressions, substitute carefully and simplify algebraically before evaluating numerical values.

Step 1: Calculate \(4*5\). \[ 4*5 = \sqrt{20} +\frac1{\sqrt{20}} \]

Step 2: Calculate \(4\odot5\). \[ 4\odot5 = \sqrt{20} -\frac1{\sqrt{20}} \]

Step 3: Evaluate the numerator. \[ (4*5)+(4\odot5) \] \[ = \left(\sqrt{20}+\frac1{\sqrt{20}}\right) + \left(\sqrt{20}-\frac1{\sqrt{20}}\right) \] The reciprocal terms cancel. \[ =2\sqrt{20} \]

Step 4: Evaluate the denominator. \[ (4*5)-(4\odot5) \] \[ = \left(\sqrt{20}+\frac1{\sqrt{20}}\right) - \left(\sqrt{20}-\frac1{\sqrt{20}}\right) \] \[ =\frac2{\sqrt{20}} \]

Step 5: Find the ratio. \[ \frac{2\sqrt{20}} {\frac2{\sqrt{20}}} \] \[ = 2\sqrt{20}\times\frac{\sqrt{20}}2 \] \[ =20 \] Therefore, \[ \boxed{20} \]