Question

Решить уравнение: (25^3x-3)^2=27^x:3^3.

Answer 1

Решение задания приложено

Answer 2

To solve the equation (25^(3x - 3))^2 = 27^(x / 3^3), we'll follow these steps:

Step 1: Simplify the exponents on both sides of the equation. Step 2: Convert all bases to the same base (either 27 or 25) on both sides of the equation. Step 3: Solve the simplified equation.

Let's start with Step 1:

Step 1: Simplify the exponents

On the left side: (25^(3x - 3))^2 = 25^(2(3x - 3)) = 25^(6x - 6)

On the right side: 27^(x / 3^3) = 27^(x / 27) = 27^x

Now, the equation becomes: 25^(6x - 6) = 27^x

Step 2: Convert bases to the same base

Since 27 is equal to 3^3, we can rewrite 27^x as (3^3)^x = 3^(3x).

Now, the equation becomes: 25^(6x - 6) = 3^(3x)

Step 3: Solve the simplified equation

To solve this equation, we need to take the logarithm of both sides. Let's take the natural logarithm (ln) for simplicity:

ln(25^(6x - 6)) = ln(3^(3x))

Using the logarithm property log(a^b) = b * log(a):

(6x - 6) * ln(25) = 3x * ln(3)

Now, isolate the variable x:

6x * ln(25) - 6 * ln(25) = 3x * ln(3)

Move all terms with x to one side:

6x * ln(25) - 3x * ln(3) = 6 * ln(25)

Factor out x:

x(6 * ln(25) - 3 * ln(3)) = 6 * ln(25)

Finally, solve for x:

x = (6 * ln(25)) / (6 * ln(25) - 3 * ln(3))

Now you can use a calculator to find the numerical value of x.