Find the solutions of the equation

3^x = x^4,
rounded to two decimal places. (Enter your answers as a comma-separated list.)

-0.80, 1.52, 7.17

x=0

Oh boy, you've stumbled onto a tricky equation! Let's clown around and find those solutions, shall we?

After doing some calculations and juggling around with numbers, I've found that the solutions to the equation 3^x = x^4 are approximately x = 0.42 and x = 5.15.

Now don't blame me if these solutions don't put a smile on your face, as they're rounded to two decimal places! So, there you have it – the solution circus has come to an end with x = 0.42 and x = 5.15.

To find the solutions of the equation 3^x = x^4, we need to solve for x. However, there is no straightforward algebraic method to obtain the exact solutions to this equation. Therefore, we will use a numerical method to approximate the solutions.

One numerical method that can be used in this case is the graphical method. We can plot the graphs of the functions f(x) = 3^x and g(x) = x^4 and find the points where the graphs intersect. These points will give us approximate values for the solutions.

To solve the equation using a graphing tool or software:
1. Plot the function f(x) = 3^x and the function g(x) = x^4 on the same coordinate system.
2. Look for the points where the two graphs intersect. These points will be the approximate solutions to the equation.
3. Round the solutions to two decimal places.

Alternatively, we can use an iterative method such as the bisection method or Newton's method to approximate the solutions. These methods involve repeatedly refining an initial guess until we get closer and closer to the true solution.

To solve the equation using an iterative method:
1. Choose an initial guess, let's say x = 1.
2. Use the chosen method to refine the guess and obtain a better approximation for the solution.
3. Repeat step 2 until you have achieved the desired level of accuracy (in this case, rounded to two decimal places).

Unfortunately, without using numerical methods or a computer program, it is not possible to obtain exact solutions for this equation.

using your favorite iterative or graphical method, you can easily find that the solutions are

-8.80, 1.52, 7.17