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31 Mar 2022, 02:30
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
45%
(medium)
Question Stats:
71% (02:00) correct
29%
(02:32)
wrong
based on 180
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\(\sqrt{4\sqrt{80} + \frac{4}{9 + 4\sqrt{5}}} = ?\)
A. \(\frac{4\sqrt{5}}{9}\)
B. 2
C. 6
D. \(\frac{4\sqrt{5}+2}{3 + 2\sqrt{5}}\)
E. 36
A. \(\frac{4\sqrt{5}}{9}\)
B. 2
C. 6
D. \(\frac{4\sqrt{5}+2}{3 + 2\sqrt{5}}\)
E. 36
31 Mar 2022, 06:30
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Bunuel
\(\sqrt{4\sqrt{80} + \frac{4}{9 + 4\sqrt{5}}} = ?\)
A. \(\frac{4\sqrt{5}}{9}\)
B. 2
C. 6
D. \(\frac{4\sqrt{5}+2}{3 + 2\sqrt{5}}\)
E. 36
A. \(\frac{4\sqrt{5}}{9}\)
B. 2
C. 6
D. \(\frac{4\sqrt{5}+2}{3 + 2\sqrt{5}}\)
E. 36
Great question!
Given: \(\sqrt{4\sqrt{80} + \frac{4}{9 + 4\sqrt{5}}}\)
Simplify the first term: \(\sqrt{4\sqrt{16}\sqrt{5} + \frac{4}{9 + 4\sqrt{5}}} = \sqrt{16\sqrt{5} + \frac{4}{9 + 4\sqrt{5}}}\)
Now let's deal with the fractional term. We can create an equivalent fraction by multiplying top and bottom by \(9 - 4\sqrt{5}\), which is the complement of \(9 + 4\sqrt{5}\)
We get: \(\sqrt{16\sqrt{5} + \frac{4(9 - 4\sqrt{5})}{(9 + 4\sqrt{5})(9 - 4\sqrt{5})}} = \sqrt{16\sqrt{5} + \frac{36 - 16\sqrt{5}}{1}}= \sqrt{16\sqrt{5} + 36 - 16\sqrt{5}} = \sqrt{36} = 6\)
Answer: C
13 Jan 2025, 06:03
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↧↧↧ Weekly Video Solution to the Problem Series ↧↧↧
We need to simplify \(\sqrt{4\sqrt{80} + \frac{4}{9 + 4\sqrt{5}}}\)
\(4\sqrt{80}\) = \(4\sqrt{16*5}\) = \(4\sqrt{4^2*5}\) = \(4 * 4\sqrt{5}\) = \(16\sqrt{5}\)
To Simplify \(\frac{4}{9 + 4\sqrt{5}}\) , We will simplify the denominator by multiply the numerator and the denominator with its conjugate \(9 - 4\sqrt{5}\)
=> \(\frac{4}{9 + 4\sqrt{5}}\) * \(\frac{ 9 - 4\sqrt{5} }{ 9 - 4\sqrt{5} }\)
=> \(\frac{ 4 * ( 9 - 4\sqrt{5} ) }{( 9 + 4\sqrt{5}) * (9 - 4\sqrt{5}) }\)
Now the denominator becomes (a-b) * (a+b) and will be equal to \(a^2 - b^2\)
=> \(\frac{ 4 * ( 9 - 4\sqrt{5} ) }{( 9^2 - (4\sqrt{5})^2)}\)
= \(\frac{36 - 16*\sqrt{5}}{81 - 16*5}\)
= \(\frac{36 - 16*\sqrt{5}}{81 - 80}\)
= \( 36 - 16*\sqrt{5} \)
=> \(\sqrt{4\sqrt{80} + \frac{4}{9 + 4\sqrt{5}}}\) = \(\sqrt{ 16\sqrt{5} + 36 - 16*\sqrt{5} }\)
= \(\sqrt{36}\) = 6
So, Answer will C
Hope it helps!
Watch the following video to learn How to Rationalize Roots
We need to simplify \(\sqrt{4\sqrt{80} + \frac{4}{9 + 4\sqrt{5}}}\)
\(4\sqrt{80}\) = \(4\sqrt{16*5}\) = \(4\sqrt{4^2*5}\) = \(4 * 4\sqrt{5}\) = \(16\sqrt{5}\)
To Simplify \(\frac{4}{9 + 4\sqrt{5}}\) , We will simplify the denominator by multiply the numerator and the denominator with its conjugate \(9 - 4\sqrt{5}\)
=> \(\frac{4}{9 + 4\sqrt{5}}\) * \(\frac{ 9 - 4\sqrt{5} }{ 9 - 4\sqrt{5} }\)
=> \(\frac{ 4 * ( 9 - 4\sqrt{5} ) }{( 9 + 4\sqrt{5}) * (9 - 4\sqrt{5}) }\)
Now the denominator becomes (a-b) * (a+b) and will be equal to \(a^2 - b^2\)
=> \(\frac{ 4 * ( 9 - 4\sqrt{5} ) }{( 9^2 - (4\sqrt{5})^2)}\)
= \(\frac{36 - 16*\sqrt{5}}{81 - 16*5}\)
= \(\frac{36 - 16*\sqrt{5}}{81 - 80}\)
= \( 36 - 16*\sqrt{5} \)
=> \(\sqrt{4\sqrt{80} + \frac{4}{9 + 4\sqrt{5}}}\) = \(\sqrt{ 16\sqrt{5} + 36 - 16*\sqrt{5} }\)
= \(\sqrt{36}\) = 6
So, Answer will C
Hope it helps!
Watch the following video to learn How to Rationalize Roots
