Question

4^sinx+2^(5-2sinx)=18

Answer 1

4^sinx + 2^5/ 2^2sinx =18

4^sinx + 32/ 4^sinx = 18 умножить всю строчку на 4^sinx

Получим: 4^2sinx +32=18* 4^ sinx

Обозначим 4^sinx за r

Получим: r^2+32=18r

Приведём запись к квадратному уравнению   r^2-18r+32=0

                                                                                       D=196=14²

                                                                                       r₁=2

                                                                                       r₂=16

возвращаемся к 4^sinx=r:

 

                               1) 4^sinx=2

                                   4^sinx=4^1/2

                                    sinx = 1/2

                                   x=(-1)^k * П/6 + Пk, k∈z.

 

                                2) 4^sinx=16

                                    4^sinx=4^2

                                     sinx=2

                                    решения нет, т.к. значение sin  в пределах от -1 до 1

 

Answer 2

To solve the equation 4^(sin(x)) + 2^(5 - 2sin(x)) = 18, we can follow these steps:

Let's start by simplifying the equation as much as possible.

First, notice that 18 can be written as 2 * 3^2. So, we can rewrite the equation as:

4^(sin(x)) + 2^(5 - 2sin(x)) = 2 * 3^2

Now, let's examine the two terms separately.

The first term, 4^(sin(x)), can be rewritten as (2^2)^(sin(x)), which simplifies to 2^(2sin(x)).

The second term, 2^(5 - 2sin(x)), remains as it is.

Substituting these simplified terms back into the equation, we have:

2^(2sin(x)) + 2^(5 - 2sin(x)) = 2 * 3^2

Next, notice that we have a common base of 2 for both terms. We can combine the terms by setting the exponents equal to each other:

2sin(x) = 5 - 2sin(x)

Now, let's solve this equation for sin(x):

2sin(x) + 2sin(x) = 5

4sin(x) = 5

sin(x) = 5/4

However, we need to check whether this solution satisfies the original equation. Substituting sin(x) = 5/4 back into the original equation:

4^(5/4) + 2^(5 - 2(5/4)) = 18

Taking the square root of both sides:

2^(5/2) + 2^(5/2) = 18

2 * (2^(5/2)) = 18

2^(7/2) = 18

This equation is not true, which means sin(x) = 5/4 is not a valid solution.

Therefore, the original equation 4^(sin(x)) + 2^(5 - 2sin(x)) = 18 does not have a real solution.