Consider This Equation:$ (4x)^{\frac{1}{4}} - X = 0 $The First Step In Solving This Equation Is To ______. The Second Step Is To ______. Solving This Equation For $ X $ Initially Yields ______. Checking The Solutions Shows That ______.

Solving the Equation: $(4x)^{\frac{1}{4}} - x = 0$

Introduction

The given equation is a simple yet challenging problem in algebra. It involves a fractional exponent and a linear term. In this article, we will guide you through the steps to solve this equation and provide a detailed explanation of each step.

Step 1: Isolate the Fractional Exponent

The first step in solving this equation is to isolate the fractional exponent. To do this, we can start by isolating the term with the fractional exponent on one side of the equation. We can do this by adding $x$ to both sides of the equation:

$$(4x)^{\frac{1}{4}} - x + x = 0 + x$$

This simplifies to:

$$(4x)^{\frac{1}{4}} = x$$

Step 2: Eliminate the Fractional Exponent

The next step is to eliminate the fractional exponent. To do this, we can raise both sides of the equation to the power of 4:

(4x)^{\frac{1}{4}}^4 = x^4

This simplifies to:

$$4x = x^4$$

Step 3: Rearrange the Equation

Now that we have eliminated the fractional exponent, we can rearrange the equation to get all the terms on one side:

$$x^4 - 4x = 0$$

Step 4: Factor the Equation

The next step is to factor the equation. We can factor out an $x$ from both terms:

$$x(x^3 - 4) = 0$$

Step 5: Solve for $x$

Now that we have factored the equation, we can solve for $x$. We can set each factor equal to zero and solve for $x$:

$$x = 0 \quad \text{or} \quad x^3 - 4 = 0$$

Solving the second equation for $x$, we get:

$$x^3 = 4 \quad \text{or} \quad x = \sqrt[3]{4}$$

Step 6: Check the Solutions

Now that we have found the solutions, we need to check them to make sure they are valid. We can do this by plugging each solution back into the original equation:

$$x = 0 \quad \text{or} \quad x = \sqrt[3]{4}$$

Plugging $x = 0$ into the original equation, we get:

$$(4(0))^{\frac{1}{4}} - 0 = 0$$

This is true, so $x = 0$ is a valid solution.

Plugging $x = \sqrt[3]{4}$ into the original equation, we get:

$$(4(\sqrt[3]{4}))^{\frac{1}{4}} - \sqrt[3]{4} = 0$$

This is also true, so $x = \sqrt[3]{4}$ is a valid solution.

Conclusion

In this article, we have solved the equation $(4x)^{\frac{1}{4}} - x = 0$ using a step-by-step approach. We have isolated the fractional exponent, eliminated the fractional exponent, rearranged the equation, factored the equation, and solved for $x$. We have also checked the solutions to make sure they are valid. The solutions to the equation are $x = 0$ and $x = \sqrt[3]{4}$.

Final Answer

The final answer is $\boxed{0, \sqrt[3]{4}}$.

Discussion

This equation is a simple yet challenging problem in algebra. It involves a fractional exponent and a linear term. The steps to solve this equation are:

1. Isolate the fractional exponent
2. Eliminate the fractional exponent
3. Rearrange the equation
4. Factor the equation
5. Solve for $x$
6. Check the solutions

These steps can be applied to solve other equations with fractional exponents and linear terms.

Related Topics

• Solving equations with fractional exponents
• Factoring equations
• Solving equations with linear terms
• Checking solutions

References

• [1] Algebra textbook, chapter 5
• [2] Online resource, solving equations with fractional exponents
• [3] Online resource, factoring equations

Keywords

• Solving equations with fractional exponents
• Factoring equations
• Solving equations with linear terms
• Checking solutions
• Algebra
• Mathematics
  Q&A: Solving the Equation $(4x)^{\frac{1}{4}} - x = 0$

Introduction

In our previous article, we solved the equation $(4x)^{\frac{1}{4}} - x = 0$ using a step-by-step approach. In this article, we will answer some frequently asked questions about solving this equation.

Q: What is the first step in solving the equation $(4x)^{\frac{1}{4}} - x = 0$?

A: The first step in solving the equation is to isolate the fractional exponent. To do this, we can add $x$ to both sides of the equation.

Q: How do I eliminate the fractional exponent?

A: To eliminate the fractional exponent, we can raise both sides of the equation to the power of 4.

Q: What is the next step after eliminating the fractional exponent?

A: After eliminating the fractional exponent, we can rearrange the equation to get all the terms on one side.

Q: How do I factor the equation?

A: We can factor the equation by factoring out an $x$ from both terms.

Q: What are the solutions to the equation?

A: The solutions to the equation are $x = 0$ and $x = \sqrt[3]{4}$.

Q: How do I check the solutions?

A: We can check the solutions by plugging each solution back into the original equation.

Q: What if I get stuck while solving the equation?

A: If you get stuck while solving the equation, you can try breaking it down into smaller steps or seeking help from a teacher or tutor.

Q: Can I use this method to solve other equations with fractional exponents?

A: Yes, you can use this method to solve other equations with fractional exponents. However, you may need to adjust the steps depending on the specific equation.

Q: What are some common mistakes to avoid while solving the equation?

A: Some common mistakes to avoid while solving the equation include:

• Not isolating the fractional exponent
• Not eliminating the fractional exponent
• Not rearranging the equation
• Not factoring the equation
• Not checking the solutions

Q: How can I practice solving equations with fractional exponents?

A: You can practice solving equations with fractional exponents by working through examples and exercises in your algebra textbook or online resources.

Q: What are some online resources for learning about solving equations with fractional exponents?

A: Some online resources for learning about solving equations with fractional exponents include:

• Khan Academy: Solving Equations with Fractional Exponents
• Mathway: Solving Equations with Fractional Exponents
• Algebra.com: Solving Equations with Fractional Exponents

Conclusion

In this article, we have answered some frequently asked questions about solving the equation $(4x)^{\frac{1}{4}} - x = 0$. We have provided step-by-step instructions and tips for solving this equation and other equations with fractional exponents.

Final Answer

The final answer is $\boxed{0, \sqrt[3]{4}}$.

Discussion

Solving equations with fractional exponents can be challenging, but with practice and patience, you can master this skill. Remember to isolate the fractional exponent, eliminate the fractional exponent, rearrange the equation, factor the equation, and check the solutions.

Related Topics

• Solving equations with fractional exponents
• Factoring equations
• Solving equations with linear terms
• Checking solutions
• Algebra
• Mathematics

References

• [1] Algebra textbook, chapter 5
• [2] Online resource, solving equations with fractional exponents
• [3] Online resource, factoring equations

Keywords

• Solving equations with fractional exponents
• Factoring equations
• Solving equations with linear terms
• Checking solutions
• Algebra
• Mathematics