Solve The Equation: $ (2x - 1)^{1/3} - 4 = 1 $

Introduction

In this article, we will delve into the world of mathematics and solve a seemingly complex equation. The equation in question is $(2x - 1)^{1/3} - 4 = 1$. This equation involves a cube root and a linear term, making it a challenging problem to solve. However, with a step-by-step approach and a solid understanding of algebraic manipulations, we can break down the equation and find the solution.

Understanding the Equation

Before we begin solving the equation, let's take a closer look at its components. The equation is $(2x - 1)^{1/3} - 4 = 1$. The left-hand side of the equation involves a cube root, which is a mathematical operation that finds the value that, when multiplied by itself twice, gives the original number. In this case, the cube root is applied to the expression $2x - 1$. The right-hand side of the equation is a simple linear term, $-4 = 1$.

Step 1: Isolate the Cube Root

To solve the equation, we need to isolate the cube root term. We can do this by adding $4$ to both sides of the equation. This will give us:

$(2x - 1)^{1/3} = 1 + 4$

Simplifying the right-hand side, we get:

$(2x - 1)^{1/3} = 5$

Step 2: Eliminate the Cube Root

Now that we have isolated the cube root term, we can eliminate it by cubing both sides of the equation. This will give us:

$(2x - 1) = 5^3$

Simplifying the right-hand side, we get:

$(2x - 1) = 125$

Step 3: Solve for x

Now that we have eliminated the cube root term, we can solve for $x$. We can do this by adding $1$ to both sides of the equation, which gives us:

$2x = 125 + 1$

Simplifying the right-hand side, we get:

$2x = 126$

Step 4: Divide by 2

Finally, we can solve for $x$ by dividing both sides of the equation by $2$. This gives us:

$x = \frac{126}{2}$

Simplifying the right-hand side, we get:

$x = 63$

Conclusion

In this article, we have solved the equation $(2x - 1)^{1/3} - 4 = 1$ using a step-by-step approach. We isolated the cube root term, eliminated it by cubing both sides of the equation, and finally solved for $x$. The solution to the equation is $x = 63$. This demonstrates the power of algebraic manipulations in solving complex equations.

Tips and Tricks

• When solving equations involving cube roots, it's essential to isolate the cube root term first.
• Cubing both sides of the equation can help eliminate the cube root term.
• Always simplify the right-hand side of the equation to make it easier to solve for the variable.

Real-World Applications

Solving equations involving cube roots has numerous real-world applications. For example, in physics, the cube root is used to calculate the volume of a cube. In engineering, the cube root is used to calculate the length of a cube's edge. In finance, the cube root is used to calculate the return on investment (ROI) of a portfolio.

Common Mistakes

When solving equations involving cube roots, it's essential to avoid common mistakes. Some common mistakes include:

• Not isolating the cube root term first.
• Not cubing both sides of the equation to eliminate the cube root term.
• Not simplifying the right-hand side of the equation.

By avoiding these common mistakes, you can ensure that you solve the equation correctly and accurately.

Conclusion

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Introduction

In our previous article, we solved the equation $(2x - 1)^{1/3} - 4 = 1$ using a step-by-step approach. However, we understand that some readers may still have questions about the solution. In this article, we will address some of the most frequently asked questions about solving the equation.

Q: What is the cube root and how is it used in the equation?

A: The cube root is a mathematical operation that finds the value that, when multiplied by itself twice, gives the original number. In the equation $(2x - 1)^{1/3} - 4 = 1$, the cube root is applied to the expression $2x - 1$. The cube root is used to isolate the variable $x$.

Q: Why do we need to isolate the cube root term first?

A: We need to isolate the cube root term first because it is a complex operation that involves multiple steps. By isolating the cube root term, we can eliminate it by cubing both sides of the equation, which makes it easier to solve for the variable $x$.

Q: What is the difference between cubing and squaring?

A: Cubing and squaring are both mathematical operations that involve raising a number to a power. However, cubing involves raising a number to the power of 3, while squaring involves raising a number to the power of 2. In the equation $(2x - 1)^{1/3} - 4 = 1$, we need to cube both sides of the equation to eliminate the cube root term.

Q: Why do we need to simplify the right-hand side of the equation?

A: We need to simplify the right-hand side of the equation to make it easier to solve for the variable $x$. Simplifying the right-hand side involves combining like terms and eliminating any unnecessary operations.

Q: What are some common mistakes to avoid when solving equations involving cube roots?

A: Some common mistakes to avoid when solving equations involving cube roots include:

• Not isolating the cube root term first
• Not cubing both sides of the equation to eliminate the cube root term
• Not simplifying the right-hand side of the equation

Q: How can I apply the steps to solve other equations involving cube roots?

A: To apply the steps to solve other equations involving cube roots, follow these steps:

1. Isolate the cube root term
2. Cube both sides of the equation to eliminate the cube root term
3. Simplify the right-hand side of the equation
4. Solve for the variable

Q: What are some real-world applications of solving equations involving cube roots?

A: Solving equations involving cube roots has numerous real-world applications, including:

• Calculating the volume of a cube in physics
• Calculating the length of a cube's edge in engineering
• Calculating the return on investment (ROI) of a portfolio in finance

Conclusion

In this article, we have addressed some of the most frequently asked questions about solving the equation $(2x - 1)^{1/3} - 4 = 1$. We hope that this Q&A guide has provided you with a better understanding of the solution and how to apply the steps to solve other equations involving cube roots.