How to Solve an Equation by Isolating the Variable x Step by Step

What We're Solving:

We need to find the value(s) of x that make the equation -1/3 x⁴ = 27 true.

The Approach:

Our goal is to isolate x by "undoing" everything that's being done to it. Right now, x is being raised to the 4th power AND multiplied by -1/3. We'll work backwards - first undoing the multiplication, then undoing the exponent. Think of it like unwrapping a present - we remove one layer at a time!

Step-by-Step Solution:

Step 1: Get rid of the fraction coefficient Starting with: -1/3 x⁴ = 27

To eliminate -1/3, we multiply both sides by -3 (the reciprocal of -1/3): (-3) × (-1/3 x⁴) = (-3) × 27 x⁴ = -81

Why this works: Multiplying by -3 cancels out the -1/3 on the left side!

Step 2: Solve for x by taking the fourth root Now we have: x⁴ = -81

To undo the 4th power, we take the 4th root of both sides: ⁴√(x⁴) = ⁴√(-81)

Step 3: Analyze what's possible Here's where we need to think carefully! We're looking for the 4th root of a negative number (-81).

When we raise any real number to an even power (like 4), we always get a positive result. Since (-81) is negative, there are no real solutions to this equation.

The Answer:

This equation has no real solutions because no real number raised to the 4th power can equal -81.

Memory Tip:

Remember: "Even powers are always positive!" When you have an even exponent (2, 4, 6, etc.) and the result is negative, there won't be any real solutions. It's like trying to find a number that, when squared, gives you -4 - impossible with real numbers!

Great job working through this tricky problem - recognizing when there's no solution is just as important as finding solutions! 🌟