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22 Apr 2020, 12:38
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
35%
(medium)
Question Stats:
71% (02:14) correct
29%
(02:45)
wrong
based on 135
sessions
History
Date
Time
Result
Not Attempted Yet
If \((25^{11}+16^{5})^{2} \)\( - (25^{11}-16^{5})^{2}=10^{n}\), what is the value of n?
(A) 11
(B) 22
(C) 33
(D) 44
(E) 55
(A) 11
(B) 22
(C) 33
(D) 44
(E) 55
23 Apr 2020, 00:21
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Ravixxx
The structure of the term is \(a^2-b^2\), where \(a^2=(25^{11}+16^{5})^{2} \) and \( b^2=(25^{11}-16^{5})^{2}\)..
Now \(a^2-b^2=(a-b)(a+b)\)
So \((25^{11}+16^{5})^{2} \)\( - (25^{11}-16^{5})^{2}=(25^{11}+16^{5}+25^{11}-16^{5})(25^{11}+16^{5}-25^{11}+16^{5})=2*25^{11}*2*16^{5}=5^{22}*2^{5*4+1+1}=10^{22}=10^{n}\)
n=22
30 Apr 2020, 01:20
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Don’t get overwhelmed by the size of the numbers here, \(25^{11}\) and \(16^5\) are easy to deal with if you can break the expression down in a way that you have to deal with these numbers only.
What makes this problem interesting is the fact that the expressions look intimidating because of the exponents. Don’t get intimidated. The expressions given are well known algebraic identities.
If we take \(25^{11}\) as A and \(16^5\) as B, the expressions automatically become \((A+B)^2\) – \((A-B)^2\) = \(10^n\).
\((a+b)^2\) – \((a-b)^2\) = (\(a^2\) + \(b^2\) + 2ab) – (\(a^2\)+\(b^2\)-2ab) = 4ab.
Therefore, \((25^{11} + 16^5)^2\) - \((25^{11} - 16^5)^2\) = 4 * \(25^{11}\) * \(16^5\) = \(2^2\) * \(5^{22}\) * \((2^4)^5\) = \(2^2\) * \(5^{22}\) * \(2^{20}\) = \(2^{22}\) * \(5^{22}\) = \(10^{22}\).
\((25^{11} + 16^5)^2\) -\( (25^{11} - 16^5)^2\) = \(10^n\)
Therefore, \(10^{22}\) = \(10^n\). Since bases are same and the numbers are equal, the exponents have to be equal. Hence, n = 22.
The correct answer option is B.
Hope that helps!
What makes this problem interesting is the fact that the expressions look intimidating because of the exponents. Don’t get intimidated. The expressions given are well known algebraic identities.
If we take \(25^{11}\) as A and \(16^5\) as B, the expressions automatically become \((A+B)^2\) – \((A-B)^2\) = \(10^n\).
\((a+b)^2\) – \((a-b)^2\) = (\(a^2\) + \(b^2\) + 2ab) – (\(a^2\)+\(b^2\)-2ab) = 4ab.
Therefore, \((25^{11} + 16^5)^2\) - \((25^{11} - 16^5)^2\) = 4 * \(25^{11}\) * \(16^5\) = \(2^2\) * \(5^{22}\) * \((2^4)^5\) = \(2^2\) * \(5^{22}\) * \(2^{20}\) = \(2^{22}\) * \(5^{22}\) = \(10^{22}\).
\((25^{11} + 16^5)^2\) -\( (25^{11} - 16^5)^2\) = \(10^n\)
Therefore, \(10^{22}\) = \(10^n\). Since bases are same and the numbers are equal, the exponents have to be equal. Hence, n = 22.
The correct answer option is B.
Hope that helps!
