# If $a = \;\sqrt {135 + \sqrt {74 + \sqrt {43 + \sqrt {36} } } }$ and $b = \;\sqrt {70 - \sqrt {41 - \sqrt {22 + \sqrt 9 } } }$ , then find the r

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If $a = \;\sqrt {135 + \sqrt {74 + \sqrt {43 + \sqrt {36} } } }$ and $b = \;\sqrt {70 - \sqrt {41 - \sqrt {22 + \sqrt 9 } } }$ , then find the ratio (a + b) ∶ (a - b).
1. 1 ∶ 5
2. 10 : 3
3. 5 ∶ 1
4. 3 : 10

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Correct Answer - Option 3 : 5 ∶ 1

Given:

$a = \;\sqrt {135 + \sqrt {74 + \sqrt {43 + \sqrt {36} } } }$

and, $b = \;\sqrt {70 - \sqrt {41 - \sqrt {22 + \sqrt 9 } } }$

Calculation:

We have, $a = \sqrt {135 + \sqrt {74 + \sqrt {43 + \sqrt {36} } } }$

$⇒ a = \sqrt {135 + \sqrt {74 + \sqrt {43 + 6} } }$

$⇒ a = \sqrt {135 + \sqrt {74 + \sqrt {49} } }$

$⇒ a = \sqrt {135 + \sqrt {74 + 7} }$

$⇒ a = \sqrt {135 + \sqrt {81} }$

$⇒ a = \sqrt {135 + 9}$

$⇒ a = \sqrt {144}$

$⇒ a = 12$

and, $b = \;\sqrt {70 - \sqrt {41 - \sqrt {22 + \sqrt 9 } } }$

$⇒ b = \;\sqrt {70 - \sqrt {41 - \sqrt {22 + 3} } }$

$⇒ b = \;\sqrt {70 - \sqrt {41 - \sqrt {25} } }$

$⇒ b = \;\sqrt {70 - \sqrt {41 - 5} }$

$⇒ b = \;\sqrt {70 - \sqrt {36} }$

$⇒ b = \;\sqrt {70 - 6}$

$⇒ b = \;\sqrt {64}$

$⇒ b = \;8$

Now, (a + b)/(a - b) = (12 + 8)/(12 - 8)

⇒ 20/4

⇒ 5/1

∴ The value of (a + b) : (a - b) is 5 : 1.