Question

Решить
$\sqrt{x\sqrt[5]{x}}-\sqrt[5]{x\sqrt{x}}=56$

Answer 1

ОДЗ: x≥0;

$\sqrt{x\sqrt[5]{x}}-\sqrt[5]{x\sqrt{x}}=56;\\\sqrt{x^{1,2}}-\sqrt[5]{x^{1,5}}=56;\\x^{0,3*2}-x^{0,3}-56=0;\ t=x^{0,3};\ t\geq 0;$

t²-t-56=0;

D=1+4*56=225;

t₁=(1-15)/2=-7; ∅ t≥0;

t₂=(1+15)/2=8;

$t=x^{0,3}=\sqrt[10]{x^3}=\sqrt[10]{x^3}=8=2^3;\ x=2^10;x^{0,6}-x^{0,3}-56=0;\\2^6-2^3=56;\\56=56;$

x=2¹⁰=1024;