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Algebra Level 1

Given that x = 2 k x={ 2 }^{ k } and 4 x = 2 c \sqrt { \dfrac { 4 }{ x } } ={ 2 }^{ c } , express c c in terms of k k .

− 1 2 k − 1 -\frac { 1 }{ 2 } k-1 − 1 2 k + 1 -\frac { 1 }{ 2 } k+1 1 2 k − 1 \frac { 1 }{ 2 } k-1 1 2 k + 1 \frac { 1 }{ 2 } k+1 k 2 \sqrt { \frac { k }{ 2 } }

Patryk Korczak by Patryk Korczak , 21, United Kingdom

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1 solution

Patryk Korczak
Dec 22, 2015

4 x = 2 c \sqrt { \frac { 4 }{ x } } ={ 2 }^{ c }

2 x = 2 c { \frac { 2 }{ \sqrt { x } } } ={ 2 }^{ c }

2 x − 1 2 = 2 c 2{ x }^{ -\frac { 1 }{ 2 } }={ 2 }^{ c }

2 ( 2 k ) − 1 2 = 2 c 2{ \left( { 2 }^{ k } \right) }^{ -\frac { 1 }{ 2 } }={ 2 }^{ c }

2 × 2 − 1 2 k = 2 c 2\times 2^{ -\frac { 1 }{ 2 } k }={ 2 }^{ c }

2 − 1 2 k + 1 = 2 c 2^{ -\frac { 1 }{ 2 } k+1 }={ 2 }^{ c }

∴ − 1 2 k + 1 = c \therefore { -\frac { 1 }{ 2 } k+1 }={ c }

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Moderator note:

Simple standard approach.

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