A Student Solved The Equation 2 T − 3 3 + 2 = 3 \sqrt[3]{2t-3} + 2 = 3 3 2 T − 3 ​ + 2 = 3 . Is The Solution Correct Or Incorrect? Explain.Solution Steps:1. 2 T − 3 3 + 2 = 3 \sqrt[3]{2t-3} + 2 = 3 3 2 T − 3 ​ + 2 = 3 2. 2 T − 3 3 = 1 \sqrt[3]{2t-3} = 1 3 2 T − 3 ​ = 1 3. Cubing Both Sides: $(\sqrt[3]{2t-3})^3 =

Introduction

In mathematics, solving equations is a crucial skill that requires attention to detail and a thorough understanding of mathematical concepts. In this article, we will analyze a student's solution to the equation $\sqrt[3]{2t-3} + 2 = 3$ and determine whether the solution is correct or incorrect.

The Student's Solution

The student's solution to the equation is as follows:

1. $\sqrt[3]{2t-3} + 2 = 3$
2. $\sqrt[3]{2t-3} = 1$
3. Cubing both sides: $(\sqrt[3]{2t-3})^3 = 1^3$
4. Simplifying: $2t-3 = 1$
5. Solving for t: $2t = 4$, $t = 2$

Analysis of the Student's Solution

Let's analyze the student's solution step by step to determine whether it is correct or incorrect.

Step 1: Is the equation $\sqrt[3]{2t-3} + 2 = 3$ correct?

The equation $\sqrt[3]{2t-3} + 2 = 3$ is a correct equation. The student has correctly isolated the cube root term.

Step 2: Is the equation $\sqrt[3]{2t-3} = 1$ correct?

The equation $\sqrt[3]{2t-3} = 1$ is also correct. The student has correctly subtracted 2 from both sides of the equation.

Step 3: Is cubing both sides of the equation $(\sqrt[3]{2t-3})^3 = 1^3$ correct?

Cubing both sides of the equation is a correct step. The student has correctly applied the exponent rule $(a^m)^n = a^{mn}$.

Step 4: Is the equation $2t-3 = 1$ correct?

The equation $2t-3 = 1$ is correct. The student has correctly simplified the equation.

Step 5: Is the solution $t = 2$ correct?

The solution $t = 2$ is incorrect. The student has correctly solved for t, but the solution is not correct.

Why is the Solution Incorrect?

The solution is incorrect because the student has not checked the domain of the cube root function. The cube root function is defined only for non-negative real numbers. In this case, the expression $2t-3$ can be negative, which means that the cube root function is not defined.

To find the correct solution, we need to check the domain of the cube root function and ensure that the expression $2t-3$ is non-negative.

Correct Solution

Let's re-solve the equation using the correct steps:

1. $\sqrt[3]{2t-3} + 2 = 3$
2. $\sqrt[3]{2t-3} = 1$
3. Cubing both sides: $(\sqrt[3]{2t-3})^3 = 1^3$
4. Simplifying: $2t-3 = 1$
5. Solving for t: $2t = 4$, $t = 2$

However, we need to check the domain of the cube root function. Let's substitute $t = 2$ into the expression $2t-3$:

$2(2) - 3 = 4 - 3 = 1$

The expression $2t-3$ is non-negative, which means that the cube root function is defined.

Conclusion

In conclusion, the student's solution to the equation $\sqrt[3]{2t-3} + 2 = 3$ is incorrect because the student has not checked the domain of the cube root function. The correct solution is $t = 2$, but we need to check the domain of the cube root function to ensure that the expression $2t-3$ is non-negative.

Final Answer

Q&A: Solving the Equation $\sqrt[3]{2t-3} + 2 = 3$

Q: What is the correct solution to the equation $\sqrt[3]{2t-3} + 2 = 3$? A: The correct solution to the equation is $t = 2$, but with the condition that the expression $2t-3$ is non-negative.

Q: Why is the solution $t = 2$ incorrect? A: The solution $t = 2$ is incorrect because the student has not checked the domain of the cube root function. The cube root function is defined only for non-negative real numbers.

Q: What is the domain of the cube root function? A: The domain of the cube root function is all real numbers $x$ such that $x \geq 0$.

Q: How do I check the domain of the cube root function? A: To check the domain of the cube root function, substitute the value of $t$ into the expression $2t-3$ and check if it is non-negative.

Q: What happens if the expression $2t-3$ is negative? A: If the expression $2t-3$ is negative, then the cube root function is not defined, and the solution is incorrect.

Q: How do I find the correct solution to the equation? A: To find the correct solution to the equation, you need to check the domain of the cube root function and ensure that the expression $2t-3$ is non-negative.

Q: What is the correct step to solve the equation? A: The correct step to solve the equation is to cube both sides of the equation and then simplify the resulting expression.

Q: Why do I need to cube both sides of the equation? A: You need to cube both sides of the equation to eliminate the cube root term and simplify the resulting expression.

Q: What is the final answer to the equation? A: The final answer to the equation is $\boxed{2}$, but with the condition that the expression $2t-3$ is non-negative.

Common Mistakes to Avoid

• Not checking the domain of the cube root function
• Not cubing both sides of the equation
• Not simplifying the resulting expression
• Not checking if the expression $2t-3$ is non-negative

Tips and Tricks

• Always check the domain of the cube root function before solving the equation
• Cube both sides of the equation to eliminate the cube root term
• Simplify the resulting expression to find the correct solution
• Check if the expression $2t-3$ is non-negative to ensure that the solution is correct

Conclusion

In conclusion, solving the equation $\sqrt[3]{2t-3} + 2 = 3$ requires careful attention to the domain of the cube root function and the correct steps to solve the equation. By following the correct steps and checking the domain of the cube root function, you can find the correct solution to the equation.