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# [Smart Math] Arithmetic Problem 26

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

### Problem

Simplify $\frac{\sqrt{11+4\sqrt{7}}+\sqrt{8+\sqrt{64}}}{\sqrt{11-4\sqrt{6}}}$

1. $\frac{2\sqrt{14}+12\sqrt{2}+\sqrt{21}-6\sqrt{3}}{5}$
2. $\frac{2\sqrt{14}+12\sqrt{2}-\sqrt{21}+6\sqrt{3}}{5}$
3. $\frac{2\sqrt{14}-12\sqrt{2}-\sqrt{21}+6\sqrt{3}}{5}$
4. $\frac{2\sqrt{14}-12\sqrt{2}+\sqrt{21}-6\sqrt{3}}{5}$
5. $\frac{2\sqrt{14}+12\sqrt{2}+\sqrt{21}+6\sqrt{3}}{5}$

### The Usual Method

Simplifying the numerator of the equation, we get:

$\sqrt{11+4\sqrt{7}}$ = $\sqrt{7+4+2\sqrt{7\times 4}}=\sqrt{7}+\sqrt{4}$

and,

$\sqrt{8+\sqrt{64}}$ = $\sqrt{8+8}=\sqrt{16}=4$

Hence numerator is $\sqrt{7}+\sqrt{4}+4$

Similarly, simplifying the denominator, we get:

$\sqrt{11-4\sqrt{6}}$ = $\sqrt{8+3-2\sqrt{6\times 4}}=\sqrt{8+3-2\sqrt{8\times 3}}=\sqrt{8}-\sqrt{3}$

Thus the equation now becomes:

$\frac{\sqrt{7}+\sqrt{4}+4}{\sqrt{8}-\sqrt{3}}$

This can be written as:

$\frac{\sqrt{7}+\sqrt{4}+4}{\sqrt{8}-\sqrt{3}}$ = $\frac{\sqrt{7}+\sqrt{4}+4}{\sqrt{8}-\sqrt{3}}\times \frac{\sqrt{8}+\sqrt{3}}{\sqrt{8}+\sqrt{3}}$

= $\frac{\sqrt{56}+6\sqrt{8}+\sqrt{21}+6\sqrt{3}}{8-3}$

= $\frac{2\sqrt{14}+12\sqrt{2}+\sqrt{21}+6\sqrt{3}}{5}$

(Ans: 5)

Estimated Time to arrive at the answer = 150 seconds.