Solve the Equation with Two Cube Roots: cuberoot(cuberoot(x)) = x

TL;DR

The video explains how to find the real solutions to an equation involving cube roots and provides step-by-step instructions.

Transcript

hi in this video we're going to find all real solutions to this equation so we have the cube root of the cube root of x and that is equal to x let's go ahead and work through its solution so we'll start by just cubing both sides of this equation that will get rid of one of the cube roots i'm going to go ahead and write it again down here just to ma... Read More

Key Insights

• 🫚 The video demonstrates how to solve an equation involving cube roots.
• 🙃 Cubing both sides of the equation helps simplify the expression.
• 🧑‍🏭 The equation can be factored using the difference of squares formula to find real solutions.
• ❓ Complex solutions are not considered because the focus is on finding real solutions only.

Questions & Answers

Q: How can we solve the equation involving the cube root of the cube root of x equal to x?

The equation is solved by cubing both sides of the equation and simplifying the expression. Then, the equation is factored to find real solutions.

Q: What is the purpose of cubing both sides of the equation?

Cubing both sides eliminates the outer cube root and simplifies the expression to the inner cube root of x equals x cubed, making it easier to solve.

Q: How is the equation factored to find real solutions?

By applying the difference of squares formula multiple times, the equation is factored as (x-1)(x+1)(x^2+1)(x^4+1) = 0. By setting each factor equal to zero, the real solutions x = 0, 1, and -1 are obtained.

Q: Why are complex solutions not considered in this problem?

Complex solutions, represented by plus or minus i when taking the square root of negative numbers, are not considered because the goal is to find only real solutions to the equation.

Summary & Key Takeaways

• The video discusses the equation involving the cube root of the cube root of x equal to x and provides a step-by-step solution.

• To solve the equation, the video demonstrates cubing both sides and simplifying the expression.

• After simplifying, the equation is factored to find real solutions.