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Arithmetic Smart Math

[Smart Math] Arithmetic Problem 26

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

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

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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.

Using Technique

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Simply observe that since all the terms in the numerator are positive, simplifying the numerator will also give all positive terms.

The denominator has a negative sign. However, the process of rationalization would require its conjugate to be multiplied to result in a rational number in the denominator. This conjugate multiplier will have positive terms. Hence, when it multiplies with the all positive term numerator, we will get a product in the numerator with all positive terms. The only option that has all positive terms in the numerator is option ‘5’.

(Ans: 5)

Estimated Time to arrive at the answer = 15 seconds.
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