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# [Smart Math] Arithmetic Problem 20

Here’s and example of a SMART MATH problem for 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

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

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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## One reply on “[Smart Math] Arithmetic Problem 20”

jitendrasays:

sir g i think is answer me – sign aayega because denominator is -1