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Question: What is the value of A:
[tex]\sf \: \sqrt{A + \sqrt{86 - \sqrt{25} } } = 20 \\ [/tex]
A. 339
B. 391
C. 409
D. 461
Answer:
[tex]\boxed{ \bf\:B. \: \: \: 391 \: } \\ [/tex]
Step-by-step explanation:
Given expression is
[tex]\sf \: \sqrt{A + \sqrt{86 - \sqrt{25} } } = 20 \\ [/tex]
[tex]\sf \: \sqrt{A + \sqrt{86 - \sqrt{5 \times 5} } } = 20 \\ [/tex]
[tex]\sf \: \sqrt{A + \sqrt{86 - 5 } } = 20 \\ [/tex]
[tex]\sf \: \sqrt{A + \sqrt{81 } } = 20 \\ [/tex]
[tex]\sf \: \sqrt{A + \sqrt{9 \times 9 } } = 20 \\ [/tex]
[tex]\sf \: \sqrt{A + 9 } = 20 \\ [/tex]
On squaring both sides, we get
[tex]\sf \: A + 9 = 400 \\ [/tex]
[tex]\sf \: A = 400 - 9\\ [/tex]
[tex]\implies\sf \:A = 391 \\ [/tex]
Hence,
[tex]\implies\sf \:\boxed{ \bf\:A = 391 \: } \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional Information:
[tex]\sf \: {( \sqrt{x} + \sqrt{y})}^{2} = x + y + 2 \sqrt{xy} \\ [/tex]
[tex]\sf \: {( \sqrt{x} - \sqrt{y})}^{2} = x + y - 2 \sqrt{xy} \\ [/tex]
[tex]\sf \: {( \sqrt{x} + \sqrt{y}) }^{2} + {( \sqrt{x} - \sqrt{y})}^{2} = 2(x + y) \\ [/tex]
[tex]\sf \: {( \sqrt{x} + \sqrt{y}) }^{2} - {( \sqrt{x} - \sqrt{y})}^{2} = 4 \sqrt{xy} \\ [/tex]
[tex]\sf \: \sqrt{x} \times \sqrt{y} = \sqrt{xy} \\ [/tex]
[tex]\sf \: \sqrt{x} + \sqrt{y} \: \ne \: \sqrt{x + y} \\ [/tex]
[tex]\sf \: \sqrt{x} - \sqrt{y} \: \ne \: \sqrt{x - y} \\ [/tex]
[tex]\sf \: \sqrt{ {x}^{2} + {y}^{2} } \: \ne \: x + y \\ [/tex]
[tex]\sf \: \sqrt{ {x}^{2} - {y}^{2} } \: \ne \: x - y \\ [/tex]