How do you solve for x in the equation 3x^4y^8 = 6x^7y^2?

To solve the equation 3x⁴y⁸ = 6x⁷y², we will start by simplifying each side of the equation.

First, we can divide both sides by y² (assuming y ≠ 0), which gives us:

3x⁴y⁶ = 6x⁷

Next, we can also divide both sides by 3:

x⁴y⁶ = 2x⁷

Now, we can rearrange the equation to isolate x:

x⁴y⁶ – 2x⁷ = 0

Factoring out x⁴ gives:

x⁴ (y⁶ – 2x³) = 0

From this equation, we have two cases to consider:

1. x⁴ = 0 which implies x = 0
2. y⁶ – 2x³ = 0 which we can rearrange to get y⁶ = 2x³

If we solve for x in the second case, we get:

x = (y⁶/2)^1/3

So, the solutions for x are:

• x = 0
• x = (y⁶/2)^1/3

These are the values of x that satisfy the original equation.