Select The Correct Answer.What Is The Solution To This Equation?$(x+6)^{\frac{1}{3}}-5=-2$A. 21 B. 7 C. -27 D. -139

Introduction

In mathematics, equations involving exponents and radicals can be challenging to solve. The given equation, $(x+6)^{\frac{1}{3}}-5=-2$, is a cubic equation that requires careful manipulation to isolate the variable $x$. In this article, we will guide you through the step-by-step process of solving this equation and selecting the correct answer from the options provided.

Understanding the Equation

The given equation is a cubic equation, which means it involves a cube root. The cube root of a number $x$ is denoted by $\sqrt[3]{x}$ or $x^{\frac{1}{3}}$. In this equation, the cube root of $(x+6)$ is subtracted by $5$ and equals $-2$.

Step 1: Isolate the Cube Root

To solve the equation, we need to isolate the cube root term. We can do this by adding $5$ to both sides of the equation:

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

$$\Rightarrow (x+6)^{\frac{1}{3}}=-2+5$$

$$\Rightarrow (x+6)^{\frac{1}{3}}=3$$

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:

$$\left((x+6)^{\frac{1}{3}}\right)^3=3^3$$

$$\Rightarrow x+6=27$$

Step 3: Solve for $x$

Finally, we can solve for $x$ by subtracting $6$ from both sides of the equation:

$$x+6=27$$

$$\Rightarrow x=27-6$$

$$\Rightarrow x=21$$

Conclusion

In this article, we have solved the cubic equation $(x+6)^{\frac{1}{3}}-5=-2$ step by step. We isolated the cube root term, eliminated it by cubing both sides of the equation, and finally solved for $x$. The correct answer is $\boxed{21}$.

Discussion

Now that we have solved the equation, let's discuss the process and the options provided.

• Option A: 21 - This is the correct answer, which we obtained by solving the equation step by step.
• Option B: 7 - This is not the correct answer, as we obtained $x=21$ by solving the equation.
• Option C: -27 - This is not the correct answer, as we obtained $x=21$ by solving the equation.
• Option D: -139 - This is not the correct answer, as we obtained $x=21$ by solving the equation.

Final Answer

The final answer is $\boxed{21}$.

Additional Resources

If you want to learn more about solving cubic equations or need help with other math problems, here are some additional resources:

• Khan Academy: Cubic Equations
• Mathway: Cubic Equation Solver
• Wolfram Alpha: Cubic Equation Solver

Introduction

In our previous article, we solved the cubic equation $(x+6)^{\frac{1}{3}}-5=-2$ step by step. In this article, we will address some frequently asked questions related to solving cubic equations.

Q: What is a cubic equation?

A: A cubic equation is a polynomial equation of degree three, which means it involves a cube root. The general form of a cubic equation is $ax^3+bx^2+cx+d=0$, where $a$, $b$, $c$, and $d$ are constants.

Q: How do I solve a cubic equation?

A: To solve a cubic equation, you need to isolate the cube root term and then eliminate it by cubing both sides of the equation. You can also use algebraic methods, such as factoring or using the rational root theorem.

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

A: A cube root is the inverse operation of cubing a number, while a square root is the inverse operation of squaring a number. For example, $\sqrt[3]{x}$ is the cube root of $x$, while $\sqrt{x}$ is the square root of $x$.

Q: Can I use a calculator to solve cubic equations?

A: Yes, you can use a calculator to solve cubic equations. However, it's always a good idea to understand the underlying math and be able to solve the equation by hand.

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

A: Some common mistakes to avoid when solving cubic equations include:

• Not isolating the cube root term
• Not eliminating the cube root term correctly
• Not checking for extraneous solutions
• Not using the correct algebraic methods

Q: Can I use technology to help me solve cubic equations?

A: Yes, you can use technology, such as graphing calculators or computer algebra systems, to help you solve cubic equations. However, it's always a good idea to understand the underlying math and be able to solve the equation by hand.

Q: What are some real-world applications of cubic equations?

A: Cubic equations have many real-world applications, including:

• Physics: Cubic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Cubic equations are used to design and optimize systems, such as bridges and buildings.
• Economics: Cubic equations are used to model economic systems and make predictions about future trends.

Conclusion

In this article, we have addressed some frequently asked questions related to solving cubic equations. We hope that this article has been helpful in clarifying any confusion and providing additional resources for learning.

Additional Resources

If you want to learn more about solving cubic equations or need help with other math problems, here are some additional resources:

• Khan Academy: Cubic Equations
• Mathway: Cubic Equation Solver
• Wolfram Alpha: Cubic Equation Solver

By following these resources, you can improve your math skills and become proficient in solving cubic equations.

Final Answer

The final answer is $\boxed{21}$.

Discussion

Now that we have addressed some frequently asked questions related to solving cubic equations, let's discuss the process and the options provided.

• Option A: 21 - This is the correct answer, which we obtained by solving the equation step by step.
• Option B: 7 - This is not the correct answer, as we obtained $x=21$ by solving the equation.
• Option C: -27 - This is not the correct answer, as we obtained $x=21$ by solving the equation.
• Option D: -139 - This is not the correct answer, as we obtained $x=21$ by solving the equation.

Final Thoughts

Solving cubic equations can be challenging, but with practice and patience, you can become proficient in solving these types of equations. Remember to always isolate the cube root term and eliminate it correctly, and don't be afraid to use technology to help you solve the equation.