Select The Correct Answer From Each Drop-down Menu.Consider This Equation:${ (4x)^{\frac{1}{3}} - X = 0 }$The First Step In Solving This Equation Is To [apply The Cube Root]. The Second Step Is To [isolate The Variable]. Solving This

Introduction

Equations with exponents can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will explore how to solve the equation $(4x)^{\frac{1}{3}} - x = 0$ using a step-by-step approach. We will apply the cube root and isolate the variable to find the solution.

Understanding the Equation

The given equation is $(4x)^{\frac{1}{3}} - x = 0$. To solve this equation, we need to apply the cube root and isolate the variable. Let's break down the equation and understand what it means.

• The expression $(4x)^{\frac{1}{3}}$ represents the cube root of $4x$. This means that we need to find the value of $x$ that makes the cube root of $4x$ equal to $x$.
• The equation is set equal to zero, which means that we need to find the value of $x$ that makes the expression $(4x)^{\frac{1}{3}} - x$ equal to zero.

Applying the Cube Root

The first step in solving this equation is to apply the cube root. This means that we need to find the value of $x$ that makes the cube root of $4x$ equal to $x$. To do this, we can start by isolating the cube root expression.

$$\begin{aligned} (4x)^{\frac{1}{3}} - x &= 0 \\ (4x)^{\frac{1}{3}} &= x \end{aligned}$$

Now, we can cube both sides of the equation to eliminate the cube root.

$$\begin{aligned} (4x)^{\frac{1}{3}} &= x \\ (4x)^{\frac{1}{3}} \times (4x)^{\frac{1}{3}} \times (4x)^{\frac{1}{3}} &= x \times x \times x \\ 4x &= x^3 \end{aligned}$$

Isolating the Variable

The second step in solving this equation is to isolate the variable. We can do this by dividing both sides of the equation by $x$.

$$\begin{aligned} 4x &= x^3 \\ \frac{4x}{x} &= \frac{x^3}{x} \\ 4 &= x^2 \end{aligned}$$

Now, we can take the square root of both sides of the equation to find the value of $x$.

$$\begin{aligned} 4 &= x^2 \\ \sqrt{4} &= \sqrt{x^2} \\ 2 &= x \end{aligned}$$

Conclusion

In this article, we have solved the equation $(4x)^{\frac{1}{3}} - x = 0$ using a step-by-step approach. We applied the cube root and isolated the variable to find the solution. The final answer is $x = 2$.

Example Questions

1. What is the first step in solving the equation $(4x)^{\frac{1}{3}} - x = 0$?
   • Apply the cube root
   • Isolate the variable
   • Divide both sides of the equation by $x$
   • Cube both sides of the equation
2. What is the second step in solving the equation $(4x)^{\frac{1}{3}} - x = 0$?
   • Apply the cube root
   • Isolate the variable
   • Divide both sides of the equation by $x$
   • Cube both sides of the equation
3. What is the final answer to the equation $(4x)^{\frac{1}{3}} - x = 0$?
   • $x = 1$
   • $x = 2$
   • $x = 4$
   • $x = 8$

Answer Key

1. Apply the cube root
2. Isolate the variable
3. $x = 2$

Discussion

This equation is a great example of how to apply the cube root and isolate the variable to solve an equation. The first step is to apply the cube root, and the second step is to isolate the variable. By following these steps, we can find the solution to the equation.

Related Topics

• Solving equations with exponents
• Applying the cube root
• Isolating the variable
• Solving quadratic equations

References

• Math Open Reference
• Khan Academy
• Wolfram Alpha
  Q&A: Solving Equations with Exponents =====================================

Frequently Asked Questions

Q: What is the first step in solving the equation $(4x)^{\frac{1}{3}} - x = 0$?

A: The first step in solving this equation is to apply the cube root. This means that we need to find the value of $x$ that makes the cube root of $4x$ equal to $x$.

Q: How do I apply the cube root in this equation?

A: To apply the cube root, we can start by isolating the cube root expression. We can do this by moving the $x$ term to the other side of the equation.

Q: What is the second step in solving the equation $(4x)^{\frac{1}{3}} - x = 0$?

A: The second step in solving this equation is to isolate the variable. We can do this by dividing both sides of the equation by $x$.

Q: How do I isolate the variable in this equation?

A: To isolate the variable, we can divide both sides of the equation by $x$. This will give us the value of $x$ that makes the expression $(4x)^{\frac{1}{3}} - x$ equal to zero.

Q: What is the final answer to the equation $(4x)^{\frac{1}{3}} - x = 0$?

A: The final answer to the equation $(4x)^{\frac{1}{3}} - x = 0$ is $x = 2$.

Q: Can you explain the concept of exponents in more detail?

A: Exponents are a way of representing repeated multiplication. For example, $x^2$ means $x$ multiplied by itself, and $x^3$ means $x$ multiplied by itself three times.

Q: How do I apply the exponent rule in this equation?

A: To apply the exponent rule, we can start by understanding the properties of exponents. For example, the product rule states that $x^a \times x^b = x^{a+b}$.

Q: What is the difference between a cube root and a square root?

A: A cube root is the inverse operation of cubing, while a square root is the inverse operation of squaring. In other words, the cube root of $x^3$ is $x$, and the square root of $x^2$ is $x$.

Q: Can you provide more examples of solving equations with exponents?

A: Here are a few more examples:

• $(2x)^{\frac{1}{2}} - x = 0$
• $(3x)^{\frac{1}{4}} - x = 0$
• $(5x)^{\frac{1}{6}} - x = 0$

Answer Key

1. Apply the cube root
2. Isolate the variable
3. $x = 2$
4. Exponents represent repeated multiplication
5. The product rule states that $x^a \times x^b = x^{a+b}$
6. A cube root is the inverse operation of cubing, while a square root is the inverse operation of squaring
7. $(2x)^{\frac{1}{2}} - x = 0$
8. $(3x)^{\frac{1}{4}} - x = 0$
9. $(5x)^{\frac{1}{6}} - x = 0$

Discussion

Solving equations with exponents can be challenging, but with the right approach, it can be done with ease. By applying the cube root and isolating the variable, we can find the solution to the equation. Remember to understand the properties of exponents and to apply the exponent rule to solve these types of equations.

Related Topics

• Solving equations with exponents
• Applying the cube root
• Isolating the variable
• Solving quadratic equations

References

• Math Open Reference
• Khan Academy
• Wolfram Alpha