3.
x^(1/4)+5*x^(1/8)-14=0 ОДЗ: x≥0
Пусть x^(1/8)=t>0 ⇒
t²+5t-14=0 D=81
t₁=2 t₂=-7 ∉ ⇒
x^(1/8)=2
(x^(1/8))⁸=2⁸
x=256.
x=256
Ответ: х=256.
1.
a) ∛(512/343)=∛(8³/7³)=∛(8/7)³=8/7.
b) ∛(75*45)=∛(3*25*9*5)=∛(3*5²*3²*5)=∛(3³*5³)=∛(3*5)³=15.
c) (10-√19)^(1/4)*(10+√19)^(1/4)=((10-√19)*(10+√19))^1/4)=
=(10²-(√19)²)^(1/4)=(100-19)^(1/4)=81^(1/4)=(3⁴)1/4)=3.
d) ((1/2)*(-20)^(1/5))⁵=(1/2)⁵*(-20^(1/5))⁵=(1/32)*(-20)=-20/32=-5/8=-0.625.
2.
a) (m³*n⁴/192)^(1/6)/(m¹⁵/(3*n⁸)^(1/6)=(m³*n⁴*3*n⁸/(192*m¹⁵)^(1/6)=
=(m³*n¹²/(64*m¹⁵)^(1/6)=(n¹²/(2⁶*m¹²)^(1/6)=n²/(2*m²).
b) (b⁷)^(1/6)*√b⁻³/∛((b⁸)^(1/4))=b^(7/6)*b^(-3/2)/∛(b^(8/4))=b^((7/6)-(3/2))/∛b²=
=b^(-1/3)/b^(2/3)=b^((-1/3)-(2/3))=b^(-1)=1/b.