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23 Aug 2015, 22:49
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For integers x and y, \(2^x + 2^y=2^{30}\). What is the value of \(x + y\)?
A. 30
B. 32
C. 46
D. 58
E. 64
A. 30
B. 32
C. 46
D. 58
E. 64
08 Jan 2016, 23:07
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shasadou
For integers x and y, 2^x+2^y=2^30. What is the value of x+y?
A. 30
B. 32
C. 46
D. 58
E. 64
A. 30
B. 32
C. 46
D. 58
E. 64
This is a property of exponents:
\(2^1 + 2^1 = 2^2\)
\(2^2 + 2^2 = 2^3\) (because \(2^2 * (1 + 1) = 2^2 * 2\))
Similarly, \(2^3 + 2^3 = 2^4\)
\(2^4 + 2^4 = 2^5\)
etc
So
\(2^{29} + 2^{29} = 2^{30}\)
\(x + y = 29 + 29 = 58\)
Similarly,
\(3^1 + 3^1 + 3^1 = 3^2\)
\(3^2 + 3^2 + 3^2 = 3^3\)
\(3^3 + 3^3 + 3^3 = 3^4\)
and so on...
24 Aug 2015, 21:38
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\(2^x+2^y\,=\,2^{30}\)
\(x+y\,=\,?\)
\(2^x\,(1+2^{y-x})\,=\,2^{30}\)
\(1+2^{y-x}\,=\,2^{30-x}\)
\(1\,=\,2^{30-x}-2^{y-x}\)
Difference between any two powers of 2 yield 1 if one of the powers is one and the other is zero
i.e. \(2^1\,-\,2^0\,=\,1\)
--> \(30-x\,=\,1\,\,\,\) and \(\,\,\,y-x\,=\,0\)
\(x\,=\,29\,\,\,\) and \(\,\,\,x\,=\,y\)
\(x+y\,=\,58\)
Answer D
\(x+y\,=\,?\)
\(2^x\,(1+2^{y-x})\,=\,2^{30}\)
\(1+2^{y-x}\,=\,2^{30-x}\)
\(1\,=\,2^{30-x}-2^{y-x}\)
Difference between any two powers of 2 yield 1 if one of the powers is one and the other is zero
i.e. \(2^1\,-\,2^0\,=\,1\)
--> \(30-x\,=\,1\,\,\,\) and \(\,\,\,y-x\,=\,0\)
\(x\,=\,29\,\,\,\) and \(\,\,\,x\,=\,y\)
\(x+y\,=\,58\)
Answer D
- big thank you! ).
