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Find the value of $\sqrt{ 70 + \sqrt{ 4896 } } + \sqrt{ 70 - \sqrt{ 4896 } }$.

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Answer:

$12$

Step by Step Explanation:

1. We need to find the value of $\sqrt{ 70 + \sqrt{ 4896 } } + \sqrt{ 70 - \sqrt{ 4896 } }$
   Let $a = \sqrt{ 70 + \sqrt{ 4896 } }$ and $b = \sqrt{ 70 - \sqrt{ 4896 } }$
2. $\begin{align} & a^2 + b^2 = 70 + \sqrt{ 4896 } + 70 - \sqrt{ 4896 } \\ \implies & a^2 + b^2 = 140 && \ldots (1)\\ \text{And}\\ & ab = \sqrt{ \left(70 + \sqrt{ 4896 }\right) \left(70 - \sqrt{ 4896 }\right) } \\ \implies & ab = \sqrt{ 70^2 - 4896 } \\ \implies & ab = \sqrt{ 4 } \\ \implies & ab = 2 && \ldots (2) \end{align}$
3. $\begin{align} & (a + b)^2 = a^2 + b^2 + 2ab \\ \implies & (a + b)^2 = 140 + 2 \times 2 && [\text{From eq(1) and eq(2)}]\\ \implies & (a + b)^2 = 144 \\ \implies & a + b = 12 \end{align}$
4. Hence, the value of $\sqrt{ 70 + \sqrt{ 4896 } } + \sqrt{ 70 - \sqrt{ 4896 } }$ is $12$.

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