What Is The Solution To This Equation?${ 4(\sqrt[3]{x-2}) + 1 = 9 }$A. 6 B. 10 C. 8 D. 0

Understanding the Equation

The given equation is $4(\sqrt[3]{x-2}) + 1 = 9$. To solve for the variable $x$, we need to isolate it on one side of the equation. The equation involves a cube root, which can be challenging to handle. However, with the correct steps, we can simplify the equation and find the solution.

Step 1: Subtract 1 from Both Sides

The first step is to isolate the term involving the cube root. We can do this by subtracting 1 from both sides of the equation. This gives us:

$4(\sqrt[3]{x-2}) = 8$

Step 2: Divide Both Sides by 4

Next, we need to get rid of the coefficient 4 that is being multiplied by the cube root. We can do this by dividing both sides of the equation by 4. This gives us:

$\sqrt[3]{x-2} = 2$

Step 3: Cube Both Sides

Now that we have isolated the cube root, we can cube both sides of the equation to get rid of the cube root. This gives us:

$x-2 = 8$

Step 4: Add 2 to Both Sides

Finally, we need to isolate the variable $x$ by adding 2 to both sides of the equation. This gives us:

$x = 10$

Conclusion

Therefore, the solution to the equation $4(\sqrt[3]{x-2}) + 1 = 9$ is $x = 10$. This is the value of $x$ that satisfies the equation.

Checking the Answer Choices

Let's check the answer choices to see which one matches our solution.

• A. 6: This is not the correct solution.
• B. 10: This is the correct solution.
• C. 8: This is not the correct solution.
• D. 0: This is not the correct solution.

Final Answer

The final answer is $\boxed{10}$.

Why is this Solution Correct?

This solution is correct because we followed the correct steps to isolate the variable $x$ and solve the equation. We subtracted 1 from both sides, divided both sides by 4, cubed both sides, and added 2 to both sides. Each step was necessary to simplify the equation and find the solution.

What if the Equation Had Been Different?

If the equation had been different, the steps to solve it would have been different. However, the general approach would have been the same: isolate the variable, simplify the equation, and solve for the variable.

What are Some Common Mistakes to Avoid?

Some common mistakes to avoid when solving equations like this include:

• Not following the correct order of operations
• Not isolating the variable correctly
• Not simplifying the equation correctly
• Not checking the answer choices

What are Some Tips for Solving Equations?

Some tips for solving equations like this include:

• Read the equation carefully and understand what it is asking for
• Follow the correct order of operations
• Isolate the variable correctly
• Simplify the equation correctly
• Check the answer choices

Conclusion

Solving equations like this requires careful attention to detail and a clear understanding of the steps involved. By following the correct steps and avoiding common mistakes, we can find the solution to the equation and check our answer choices.

Q: What is the first step in solving an equation like $4(\sqrt[3]{x-2}) + 1 = 9$?

A: The first step is to isolate the term involving the cube root by subtracting 1 from both sides of the equation. This gives us $4(\sqrt[3]{x-2}) = 8$.

Q: How do I get rid of the coefficient 4 that is being multiplied by the cube root?

A: To get rid of the coefficient 4, we need to divide both sides of the equation by 4. This gives us $\sqrt[3]{x-2} = 2$.

Q: What is the next step after isolating the cube root?

A: The next step is to cube both sides of the equation to get rid of the cube root. This gives us $x-2 = 8$.

Q: How do I find the value of $x$?

A: To find the value of $x$, we need to add 2 to both sides of the equation. This gives us $x = 10$.

Q: What if the equation had a different coefficient or a different operation?

A: If the equation had a different coefficient or a different operation, the steps to solve it would be different. However, the general approach would be the same: isolate the variable, simplify the equation, and solve for the variable.

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

A: Some common mistakes to avoid when solving equations include not following the correct order of operations, not isolating the variable correctly, not simplifying the equation correctly, and not checking the answer choices.

Q: How can I check my answer choices?

A: To check your answer choices, you can plug each choice into the original equation and see which one satisfies the equation.

Q: What if I get stuck on a problem?

A: If you get stuck on a problem, try breaking it down into smaller steps, or ask for help from a teacher or tutor.

Q: What are some tips for solving equations?

A: Some tips for solving equations include reading the equation carefully and understanding what it is asking for, following the correct order of operations, isolating the variable correctly, simplifying the equation correctly, and checking the answer choices.

Q: Can I use a calculator to solve equations?

A: Yes, you can use a calculator to solve equations, but make sure to check your work and understand the steps involved.

Q: What if I make a mistake on a problem?

A: If you make a mistake on a problem, don't worry! Just go back and recheck your work, and try again.

Q: How can I practice solving equations?

A: You can practice solving equations by working on problems in a textbook or online, or by taking a practice test.

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

A: Solving equations has many real-world applications, such as solving problems in physics, engineering, and economics.

Q: Can I use equations to solve problems in other subjects?

A: Yes, you can use equations to solve problems in other subjects, such as algebra, geometry, and trigonometry.

Q: What if I need help with a specific problem?

A: If you need help with a specific problem, try asking a teacher or tutor for assistance.

Q: How can I stay motivated when solving equations?

A: You can stay motivated by setting goals for yourself, rewarding yourself for completing problems, and finding a study group or partner to work with.

Q: What are some resources for learning more about solving equations?

A: Some resources for learning more about solving equations include textbooks, online tutorials, and practice problems.

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

A: Yes, you can use technology, such as calculators and computer software, to help you solve equations.

Q: What if I'm struggling with a concept?

A: If you're struggling with a concept, try breaking it down into smaller steps, or ask for help from a teacher or tutor.

Q: How can I use equations to solve real-world problems?

A: You can use equations to solve real-world problems by applying the concepts and techniques you've learned to real-world situations.

Q: What are some common types of equations?

A: Some common types of equations include linear equations, quadratic equations, and polynomial equations.

Q: How can I use equations to model real-world situations?

A: You can use equations to model real-world situations by using variables and constants to represent real-world quantities and relationships.

Q: What if I need to solve a system of equations?

A: If you need to solve a system of equations, try using substitution or elimination methods to find the solution.

Q: How can I use equations to solve optimization problems?

A: You can use equations to solve optimization problems by using variables and constants to represent real-world quantities and relationships, and then using calculus to find the maximum or minimum value.

Q: What are some common applications of equations in science and engineering?

A: Some common applications of equations in science and engineering include solving problems in physics, engineering, and economics.

Q: How can I use equations to solve problems in finance?

A: You can use equations to solve problems in finance by using variables and constants to represent real-world quantities and relationships, and then using calculus to find the maximum or minimum value.

Q: What if I need to solve a differential equation?

A: If you need to solve a differential equation, try using separation of variables or other techniques to find the solution.

Q: How can I use equations to solve problems in computer science?

A: You can use equations to solve problems in computer science by using variables and constants to represent real-world quantities and relationships, and then using calculus to find the maximum or minimum value.

Q: What are some common types of differential equations?

A: Some common types of differential equations include first-order differential equations, second-order differential equations, and higher-order differential equations.

Q: How can I use equations to solve problems in biology?

A: You can use equations to solve problems in biology by using variables and constants to represent real-world quantities and relationships, and then using calculus to find the maximum or minimum value.

Q: What if I need to solve a partial differential equation?

A: If you need to solve a partial differential equation, try using separation of variables or other techniques to find the solution.

Q: How can I use equations to solve problems in economics?

A: You can use equations to solve problems in economics by using variables and constants to represent real-world quantities and relationships, and then using calculus to find the maximum or minimum value.

Q: What are some common applications of equations in economics?

A: Some common applications of equations in economics include solving problems in macroeconomics, microeconomics, and international trade.

Q: How can I use equations to solve problems in physics?

A: You can use equations to solve problems in physics by using variables and constants to represent real-world quantities and relationships, and then using calculus to find the maximum or minimum value.

Q: What are some common types of equations in physics?

A: Some common types of equations in physics include Newton's laws of motion, the laws of thermodynamics, and the equations of motion for objects in a gravitational field.

Q: How can I use equations to solve problems in engineering?

A: You can use equations to solve problems in engineering by using variables and constants to represent real-world quantities and relationships, and then using calculus to find the maximum or minimum value.

Q: What are some common applications of equations in engineering?

A: Some common applications of equations in engineering include solving problems in mechanical engineering, electrical engineering, and civil engineering.

Q: How can I use equations to solve problems in computer science?

A: You can use equations to solve problems in computer science by using variables and constants to represent real-world quantities and relationships, and then using calculus to find the maximum or minimum value.

Q: What are some common types of equations in computer science?

A: Some common types of equations in computer science include linear equations, quadratic equations, and polynomial equations.

Q: How can I use equations to solve problems in data analysis?

A: You can use equations to solve problems in data analysis by using variables and constants to represent real-world quantities and relationships, and then using calculus to find the maximum or minimum value.

Q: What are some common applications of equations in data analysis?

A: Some common applications of equations in data analysis include solving problems in regression analysis, time series analysis, and hypothesis testing.

Q: How can I use equations to solve problems in machine learning?

A: You can use equations to solve problems in machine learning by using variables and constants to represent real-world quantities and relationships, and then using calculus to find the maximum or minimum value.

Q: What are some common types of equations in machine learning?

A: Some common types of equations in machine learning include linear equations, quadratic equations, and polynomial equations.

Q: How can I use equations to solve problems in natural language processing?

A: You can use equations to solve problems in natural language processing by using variables and constants to represent real-world quantities and relationships, and then using calculus to find the maximum or minimum value.

Q: What are some common applications of equations in natural language processing?

A: Some common applications of equations in natural language processing include solving problems in text classification, sentiment analysis, and language modeling.

Q: How can I use equations to solve problems in computer vision?

A: You can use equations to solve problems in computer vision by using variables and constants to represent real-world quantities and relationships, and then using calculus to find the maximum or minimum value.

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