1) а) x^3 - 5 = 3
x^3 = 8 = 2^3
x = 2
b) 4x^3 + 10 = 6
4x^3 = 6 - 10 = -4
x^3 = -1; x = -1
c) (x + 5)^3 = -1/8 = (-1/2)^3
x + 5 = -1/2
x = -5 1/2 = -5,5
2) a) √x + 1 = 5
√x = 4; x = 4^2 = 16
b) 4√x - 12 = 20
4√x = 20 + 12 = 32
√x = 32/4 = 8
x = 8^2 = 64
c) √(2x-4) = 6
2x - 4 = 6^2 = 36
2x = 36 + 4 = 40
x = 40/2 = 20
3) a) 3^x + 5 = 6
3^x = 6 - 5 = 1 = 3^0
x = 0
b) 23 - 2*4^x = -9
23 + 9 = 2*4^x
32 = 2*4^x
4^x = 32/2 = 16 = 4^2
x = 2
c) 5^(x^2-6x+3) = 1/25 = 5^(-2)
x^2 - 6x + 3 = -2
x^2 - 6x + 5 = 0
(x - 1)(x - 5) = 0
x1 = 1; x2 = 5
4) a) log2(x) = 5,2; x = 2^(5,2)
b) lg(5x) = lg(x^2+6)
Область определения: x > 0
5x = x^2 + 6
x^2 - 5x + 6 = 0
(x - 2)(x - 3) = 0
x1 = 2; x2 = 3
c) log2(x+4) + 3 = 1
log2(x+4) = 1 - 3 = -2
x + 4 = 2^(-2) = 1/4
x = -4 + 1/4 = -3,75
5) a) |x| = 5; x1 = -5; x2 = 5
b) 3*|x| + 4 = 7
3*|x| = 7 - 4 = 3
|x| = 1; x1 = -1; x2 = 1
c) |x+1| = 2
x + 1 = -2; x1 = -3
x + 1 = 2; x2 = 1
6) log3(x - 7) = log3(3 - x)
Область определения слева x > 7
Область определения справа x < 3
Решений нет
7) Здесь графически показано решение уравнения
√x = 1
Очевидно, что x = 1 и других решений нет.