Wurzelrechnung, nach x auflösen: √(8·x·(8·x)^{1/3}) - √(x·x^{1/3}) = 27/4

Question

$$ \sqrt { 8x\quad \times \quad \sqrt [ 3 ]{ 8x }  } -\sqrt { x\quad \times \quad \sqrt [ 3 ]{ x }  } =\frac { 27 }{ 4 }  $$

Answer 1

√(8·x·(8·x)^{1/3}) - √(x·x^{1/3}) = 27/4

√(16·x^{4/3}) - √(x^{4/3}) = 27/4

4·x^{2/3} - x^{2/3} = 27/4

3·x^{2/3} = 27/4

x^{2/3} = 9/4

x = 27/8

Answer 2

erst mal was vereinfachen:

(8*x*(8*x)^(1/3) ) ^(1/2)   -   ( x ^(4/3) ) ^(1/2) = 27/4

(  ( 8x) ^(4/3)  ) ^(1/2)     -   ^{x (2/3)}    = 27/4

( 8x) ^(2/3)    -   x ^(2/3)    = 27/4        jetzt links ausklammern

  x ^(2/3)     ( 8 (2/3)    -  1 ) =  27/4   

3* x ^(2/3)     =  27/4   

x ^(2/3)     =  9/4   

x = 27/8