What Is The Solution Of $\sqrt[3]{x+8}=-4$?A. $x = -72$ B. $x = -66$ C. $x = 58$ D. No Solution

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Introduction

When dealing with equations involving cube roots, it's essential to understand the properties of cube roots and how to isolate the variable. In this case, we're given the equation $\sqrt[3]{x+8}=-4$, and we need to find the solution for $x$. We'll start by understanding the properties of cube roots and then proceed to solve the equation.

Properties of Cube Roots

The cube root of a number is a value that, when multiplied by itself twice, gives the original number. In other words, if $a = \sqrt[3]{b}$, then $a^3 = b$. This property is crucial in solving equations involving cube roots.

Solving the Equation

To solve the equation $\sqrt[3]{x+8}=-4$, we need to isolate the variable $x$. We can start by cubing both sides of the equation, which will eliminate the cube root.

Cubing Both Sides

When we cube both sides of the equation, we get:

$$\left(\sqrt[3]{x+8}\right)^3 = (-4)^3$$

This simplifies to:

$$x+8 = -64$$

Isolating the Variable

Now, we need to isolate the variable $x$. We can do this by subtracting 8 from both sides of the equation.

$$x+8-8 = -64-8$$

This simplifies to:

$$x = -72$$

Conclusion

Therefore, the solution to the equation $\sqrt[3]{x+8}=-4$ is $x = -72$. This means that when we substitute $x = -72$ into the original equation, the equation holds true.

Discussion

It's worth noting that the equation $\sqrt[3]{x+8}=-4$ has a unique solution, which is $x = -72$. This is because the cube root function is one-to-one, meaning that each output corresponds to a unique input. Therefore, there is only one solution to the equation.

Final Answer

The final answer is $\boxed{-72}$.

Comparison of Options

Let's compare our solution with the given options:

• A. $x = -72$: This is the solution we obtained.
• B. $x = -66$: This is not the solution we obtained.
• C. $x = 58$: This is not the solution we obtained.
• D. No solution: This is incorrect, as we obtained a unique solution.

Therefore, the correct answer is A. $x = -72$.

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Introduction

In our previous article, we solved the equation $\sqrt[3]{x+8}=-4$ and obtained the solution $x = -72$. However, we received several questions from readers regarding the solution and the steps involved in solving the equation. In this article, we'll address some of the most frequently asked questions and provide additional insights into the solution.

Q: What is the cube root of a number?

A: The cube root of a number is a value that, when multiplied by itself twice, gives the original number. In other words, if $a = \sqrt[3]{b}$, then $a^3 = b$.

Q: Why did we cube both sides of the equation?

A: We cubed both sides of the equation to eliminate the cube root. By doing so, we were able to isolate the variable $x$ and solve for its value.

Q: What is the significance of the cube root function being one-to-one?

A: The cube root function is one-to-one, meaning that each output corresponds to a unique input. This is why we obtained a unique solution to the equation, which is $x = -72$.

Q: Can we use other methods to solve the equation?

A: Yes, we can use other methods to solve the equation. However, cubing both sides of the equation is a straightforward and efficient way to eliminate the cube root and solve for the variable.

Q: What if the equation had a different cube root?

A: If the equation had a different cube root, we would need to adjust our solution accordingly. For example, if the equation was $\sqrt[3]{x+8}=2$, we would need to cube both sides of the equation and solve for $x$.

Q: Can we apply this method to other equations involving cube roots?

A: Yes, we can apply this method to other equations involving cube roots. The key is to cube both sides of the equation and solve for the variable.

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 cubing both sides of the equation
• Not isolating the variable
• Not checking the solution for validity

Q: How can we check the solution for validity?

A: We can check the solution for validity by substituting the value of $x$ back into the original equation and verifying that it holds true.

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

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

• Physics: When dealing with problems involving volume and density, cube roots are often used to calculate the volume of a cube.
• Engineering: When designing structures, cube roots are used to calculate the volume of materials.
• Computer Science: When working with algorithms, cube roots are used to calculate the time complexity of a problem.

Conclusion

In conclusion, solving equations involving cube roots requires a clear understanding of the properties of cube roots and how to isolate the variable. By cubing both sides of the equation and solving for the variable, we can obtain a unique solution. We hope this Q&A article has provided additional insights into the solution and has helped readers better understand the concept.

Final Answer

The final answer is $\boxed{-72}$.

Comparison of Options

Let's compare our solution with the given options:

• A. $x = -72$: This is the solution we obtained.
• B. $x = -66$: This is not the solution we obtained.
• C. $x = 58$: This is not the solution we obtained.
• D. No solution: This is incorrect, as we obtained a unique solution.

Therefore, the correct answer is A. $x = -72$.