Select The Correct Answer.What Is The Solution To The Equation $\sqrt[3]{x-1}=-4$?A. -11 B. -63 C. -65 D. No Solution

Feb 27, 2025 by ADMIN 121 views

**Solving the Cube Root Equation: A Step-by-Step Guide**

Introduction

In this article, we will explore the solution to the equation $\sqrt[3]{x-1}=-4$ . This equation involves a cube root, which can be solved using algebraic techniques. We will break down the solution into manageable steps, making it easy to understand and follow along.

Understanding the Equation

The given equation is $\sqrt[3]{x-1}=-4$ . To solve this equation, we need to isolate the variable $x$ . The cube root can be eliminated by cubing both sides of the equation.

Step 1: Cube Both Sides

Cubing both sides of the equation $\sqrt[3]{x-1}=-4$ gives us:

\left(\sqrt[3]{x-1}\right)^3=(-4)^3

This simplifies to:

x-1=-64

Step 2: Add 1 to Both Sides

To isolate the variable $x$ , we need to add 1 to both sides of the equation:

x-1+1=-64+1

This simplifies to:

x=-63

Conclusion

The solution to the equation $\sqrt[3]{x-1}=-4$ is $x=-63$ . This is the only solution to the equation, as the cube root is a one-to-one function.

Why is there only one solution?

The cube root function is a one-to-one function, meaning that each input corresponds to a unique output. In this case, the input is $x-1$ , and the output is $\sqrt[3]{x-1}$ . Since the cube root function is one-to-one, there is only one possible value of $x$ that satisfies the equation.

What if the cube root is not a one-to-one function?

If the cube root were not a one-to-one function, there would be multiple solutions to the equation. However, this is not the case, as the cube root function is a one-to-one function.

Real-World Applications

The solution to the equation $\sqrt[3]{x-1}=-4$ has real-world applications in various fields, such as:

Physics: The equation can be used to model the motion of objects under the influence of gravity.
Engineering: The equation can be used to design and optimize systems, such as bridges and buildings.
Computer Science: The equation can be used to develop algorithms and models for solving complex problems.

Conclusion

In conclusion, the solution to the equation $\sqrt[3]{x-1}=-4$ is $x=-63$ . This is the only solution to the equation, as the cube root function is a one-to-one function. The solution has real-world applications in various fields, making it an important concept to understand.

Frequently Asked Questions

Q: What is the cube root function?

A: The cube root function is a mathematical function that takes a number as input and returns a number as output. The cube root function is denoted by $\sqrt[3]{x}$ .

Q: Why is the cube root function one-to-one?

A: The cube root function is one-to-one because each input corresponds to a unique output. This means that if two inputs are equal, their corresponding outputs will also be equal.

Q: What are some real-world applications of the cube root function?

A: The cube root function has real-world applications in various fields, such as physics, engineering, and computer science. It can be used to model the motion of objects, design and optimize systems, and develop algorithms and models for solving complex problems.

Q: How do I solve a cube root equation?

A: To solve a cube root equation, you need to cube both sides of the equation and then isolate the variable. This will give you the solution to the equation.

References

Q: What is a cube root equation?

A: A cube root equation is an equation that involves a cube root, which is a mathematical operation that finds the number that, when multiplied by itself twice, gives a specified value.

Q: How do I solve a cube root equation?

A: To solve a cube root equation, you need to cube both sides of the equation and then isolate the variable. This will give you the solution to the equation.

Q: What is the cube root function?

A: The cube root function is a mathematical function that takes a number as input and returns a number as output. The cube root function is denoted by $\sqrt[3]{x}$ .

Q: Why is the cube root function one-to-one?

A: The cube root function is one-to-one because each input corresponds to a unique output. This means that if two inputs are equal, their corresponding outputs will also be equal.

Q: What are some real-world applications of the cube root function?

Q: How do I cube both sides of an equation?

A: To cube both sides of an equation, you need to multiply both sides of the equation by itself three times. This will give you the cube of the left-hand side and the cube of the right-hand side.

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

A: A cube root is a mathematical operation that finds the number that, when multiplied by itself twice, gives a specified value. A square root is a mathematical operation that finds the number that, when multiplied by itself, gives a specified value.

Q: Can I use a calculator to solve a cube root equation?

A: Yes, you can use a calculator to solve a cube root equation. However, it's always a good idea to check your work by cubing both sides of the equation and then isolating the variable.

Q: What if I have a cube root equation with a negative number?

A: If you have a cube root equation with a negative number, you can still solve it by cubing both sides of the equation and then isolating the variable. However, you may need to use the properties of negative numbers to simplify the equation.

Q: Can I use the cube root function to solve equations with variables on both sides?

A: Yes, you can use the cube root function to solve equations with variables on both sides. However, you may need to use algebraic techniques, such as adding or subtracting the same value to both sides of the equation, to isolate the variable.

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

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

Not cubing both sides of the equation
Not isolating the variable
Not checking your work
Not using the properties of negative numbers
Not using algebraic techniques to simplify the equation

Conclusion

Solving cube root equations can be a challenging task, but with practice and patience, you can master it. Remember to cube both sides of the equation, isolate the variable, and check your work to ensure that you have the correct solution. With these tips and techniques, you'll be able to solve cube root equations with ease.