Select The Correct Answer.What Is The Solution To The Equation?$3(x+9)^{\frac{3}{4}}=24$A. -3 B. 6 C. 7 D. 25

Introduction

Equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific equation, $3(x+9)^{\frac{3}{4}}=24$, and provide a step-by-step guide to help you understand the solution.

Understanding the Equation

The given equation is $3(x+9)^{\frac{3}{4}}=24$. This equation involves an exponent, a coefficient, and a constant. To solve this equation, we need to isolate the variable $x$.

Step 1: Divide Both Sides by 3

The first step in solving this equation is to divide both sides by 3. This will help us get rid of the coefficient 3.

$$\frac{3(x+9)^{\frac{3}{4}}}{3}=\frac{24}{3}$$

$$\Rightarrow (x+9)^{\frac{3}{4}}=8$$

**Step 2: Raise Both Sides to the Power of $\frac{4}{3}$

To get rid of the exponent $\frac{3}{4}$, we need to raise both sides to the power of $\frac{4}{3}$.

$$\left((x+9)^{\frac{3}{4}}\right)^{\frac{4}{3}}=(8)^{\frac{4}{3}}$$

$$\Rightarrow x+9=16$$

Step 3: Subtract 9 from Both Sides

The final step is to subtract 9 from both sides to isolate the variable $x$.

$$x+9-9=16-9$$

$$\Rightarrow x=7$$

Conclusion

In this article, we solved the equation $3(x+9)^{\frac{3}{4}}=24$ using a step-by-step approach. We divided both sides by 3, raised both sides to the power of $\frac{4}{3}$, and finally subtracted 9 from both sides to isolate the variable $x$. The solution to the equation is $x=7$.

Answer

The correct answer is C. 7.

Why is this solution correct?

This solution is correct because we followed a logical and step-by-step approach to solve the equation. We used algebraic properties and operations to isolate the variable $x$, and the final answer satisfies the original equation.

Tips and Tricks

• When solving equations with exponents, it's essential to use the correct exponent rules.
• Make sure to isolate the variable $x$ by using inverse operations.
• Check your solution by plugging it back into the original equation.

Common Mistakes

• Failing to isolate the variable $x$.
• Using the wrong exponent rules.
• Not checking the solution by plugging it back into the original equation.

Real-World Applications

Solving equations with exponents has numerous real-world applications, such as:

• Physics: Solving equations with exponents is crucial in physics, particularly in the study of motion and energy.
• Engineering: Engineers use equations with exponents to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Economists use equations with exponents to model economic systems and make predictions about future trends.

Conclusion

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Q: What is the first step in solving an equation with an exponent?

A: The first step in solving an equation with an exponent is to isolate the variable by using inverse operations. In the case of the equation $3(x+9)^{\frac{3}{4}}=24$, the first step is to divide both sides by 3.

Q: How do I deal with exponents in equations?

A: When dealing with exponents in equations, it's essential to use the correct exponent rules. For example, when raising both sides to a power, you need to multiply the exponents. In the case of the equation $3(x+9)^{\frac{3}{4}}=24$, we raised both sides to the power of $\frac{4}{3}$.

Q: What is the difference between a coefficient and an exponent?

A: A coefficient is a number that is multiplied by a variable, while an exponent is a power to which a variable is raised. In the equation $3(x+9)^{\frac{3}{4}}=24$, 3 is the coefficient and $\frac{3}{4}$ is the exponent.

Q: How do I check my solution to an equation with an exponent?

A: To check your solution to an equation with an exponent, plug the solution back into the original equation and verify that it is true. In the case of the equation $3(x+9)^{\frac{3}{4}}=24$, we can plug in $x=7$ and verify that it satisfies the equation.

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

A: Some common mistakes to avoid when solving equations with exponents include:

• Failing to isolate the variable
• Using the wrong exponent rules
• Not checking the solution by plugging it back into the original equation

Q: How do I apply the skills I learned in this article to real-world problems?

A: The skills you learned in this article can be applied to a wide range of real-world problems, including:

• Physics: Solving equations with exponents is crucial in physics, particularly in the study of motion and energy.
• Engineering: Engineers use equations with exponents to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Economists use equations with exponents to model economic systems and make predictions about future trends.

Q: What are some additional resources I can use to learn more about solving equations with exponents?

A: Some additional resources you can use to learn more about solving equations with exponents include:

• Online tutorials and videos
• Math textbooks and workbooks
• Online forums and discussion groups

Q: How can I practice solving equations with exponents?

A: You can practice solving equations with exponents by:

• Working through practice problems in a math textbook or workbook
• Using online resources, such as Khan Academy or Mathway
• Creating your own practice problems and solving them

Conclusion

In conclusion, solving equations with exponents requires a solid understanding of algebraic properties and operations. By following the steps outlined in this article and practicing with real-world problems, you can become proficient in solving equations with exponents and apply the skills to a wide range of fields.