The Confusionist 4

Algebra Level pending

If 4 x = 1 2 4 ^ x = \frac{1}{2} , where x x is in the form − a b - \frac{a}{b} , where positive integers a a and b b are coprimes, then what is a + b a + b ?

-2 2 -1 4 1 3

Mihir Chaturvedi by Mihir Chaturvedi , 20, India

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1 solution

Mihir Chaturvedi
Jan 15, 2016

4 x = 1 2 ⇒ ( 2 ) 2 x = 2 − 1 ⇒ 2 2 x = 2 − 1 { 4 }^{ x }=\frac { 1 }{ 2 } \\ \Rightarrow { (2) }^{ { 2 }^{ x } }={ 2 }^{ -1 }\\ \Rightarrow { 2 }^{ 2x }={ 2 }^{ -1 }

Since the bases are equal, thus the exponents can be equated, 2 x = − 1 ⇒ x = − 1 2 2x=-1\\ \Rightarrow x=\frac { -1 }{ 2 }

a b = − 1 2 \frac { a }{ b } = \frac{-1}{2} , then a + b = − 1 + 2 = 3 a+b=-1+2=3

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