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Arithmetic Smart Math

[Smart Math] Arithmetic Problem 20

Here’s and example of a SMART MATH problem for ARITHMETIC.

Arithmetic

Problem

\frac{\sqrt{6}+\sqrt{7}}{\sqrt{5}-\sqrt{6}}=?

  1. 6+\sqrt{30}+\sqrt{35}+\sqrt{42}
  2. 6-\sqrt{30}+\sqrt{35}+\sqrt{42}
  3. 6+\sqrt{30}-\sqrt{35}+\sqrt{42}
  4. 6-\sqrt{30}-\sqrt{35}-\sqrt{42}
  5. 6+\sqrt{30}+\sqrt{35}-\sqrt{42}

The Usual Method

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Using Rationalization:

\frac{\sqrt{6}+\sqrt{7}}{\sqrt{5}-\sqrt{6}}= \frac{\left( \sqrt{6}+\sqrt{7} \right)}{\left( \sqrt{5}-\sqrt{6} \right)}\times \frac{\left( \sqrt{5}+\sqrt{6} \right)}{\left( \sqrt{5}+\sqrt{6} \right)}

= \frac{\sqrt{30}+6+\sqrt{35}+\sqrt{42}}{5-6}=\sqrt{30}+6+\sqrt{35}+\sqrt{42}

(Ans: 1)

Estimated Time to arrive at the answer = 30 seconds.

Using Technique

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Since the denominator is \left( \sqrt{5}-\sqrt{6} \right), its conjugate is \left( \sqrt{5}+\sqrt{6} \right)which is multiplied to it and the numerator \left( \sqrt{6}+\sqrt{7} \right). Since both the terms ( \left( \sqrt{5}+\sqrt{6} \right) and \left( \sqrt{6}+\sqrt{7} \right)) are positive terms, their product too would be positive and there would not be any negative sign. The only option that does not have a negative sign is option ‘1’.

(Ans: 1)

Estimated Time to arrive at the answer = 5 seconds.
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