1)
4^(x-2)-17*2^(x-4)+1=0
2^(2x-4)-17*2^(x-4)+1=0
2^(2x)/16-17*2^x/16+1=0 |×16
2^(2x)-17*2^x+16=0
2^x=t>0 ⇒
t²-17t+16=0 D=225
t₁=16 ⇒ 2^x=16 2^x=2⁴ x₁=4
t₂=1 ⇒ 2^x=1 2^x=2⁰ x₂=0
Ответ: x₁=4 x₂=0.
2)
log₁/₂(x-√(x²-16)=-1 ОДЗ: х²-16≥0 (х-4)(х+4)≥0 х∈(-∞;-4]U[4;+∞)
x-√(x²-16)=(1/2)⁻¹
x-√(x²-16)=2
(√(x²-16))²=(2-x)²
x²-16=4-4x+x²
4x=20
x=5 ∈ОДЗ
Ответ: х=5.
3)
2x⁴-50m²=2m²x²+50x²
(2x⁴-2m²x²)-(50x²-50m²)=0
2x²(x²-m²)-50(x²-m²)=0
(x²-m²)(2x²-50=0
x²-m²=0 x²=m²
2x²-50=0 |÷2 x²=25=m² ⇒
x ₁=m₁=5 x₂=5 m₂=-5
x₃=m ₃=-5.= x₄=-5 m₄=5.
4)
x+√(x+1)=11 ОДЗ: х+1≥0 х≥-1
(√(x+1))²=(11-x)²
x+1=121-22x+x²
x²-23x+120=0 D=49
x₁=8∈ОДЗ x₂=15∈ОДЗ.
5)
(1/16)^(-3/4)+343^(1/3)+(1/8)^(-2/3)*0,81^(-0,5)=16^(3/4)+7+8^(2/3)*(81/100)^(-0,5)=
=2^(4*3/4)+7+2^(3*2/3)*(100/81)^0,5=2³+7+2²*10/9=8+7+4*10/9=15+40/9=19⁴/₉.
6)
Упростим левую часть выражения:
((a^(1/3)-x^(1/3)⁻¹*(a-x)-(a+x)/(a^(1/3)-x^(1/3))=
((a-x)/(a^(1/3)-x^(1/3))-(a+x)/(a^(1/3)+x^(1/3))=
((a-x)(a^(1/3)+x^(1/3))-(a+x)(a^(1/3)-x^(1/3))/(a^(2/3)-x^(2/3))=
a^(4/3)-a^(1/3)x+ax^(1/3)-x^(4/3)-a^(4/3)-a^(1/3)x+ax^(1/3)+x^(4/3))/(a^(2/3)-x^(2/3))
=(2ax^(1/3)-2a^(1/3)x)/(a^(2/3)-x^(2/3))=2a^(1/3)x^(1/3)(a^(2/3)-x^(2/3)/(a^(2/3)-x^(2/3)=
=2a^(1/3)x^(1/3). ⇒
2a^(1/3)x^(1/3)*2(ax)^(-1/3)=2(a^(1/3)x^(1/3)*2/(a^1/3)x^(1/3))=4.