Solve The Equation And Check For Extraneous Solutions. X 2 3 = 9 X^{\frac{2}{3}} = 9 X 3 2 ​ = 9

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Introduction

In mathematics, solving equations is a crucial skill that helps us find the value of unknown variables. When dealing with equations involving exponents, it's essential to understand the properties of exponents and how to manipulate them to isolate the variable. In this article, we will solve the equation x23=9x^{\frac{2}{3}} = 9 and check for extraneous solutions.

Understanding the Equation

The given equation is x23=9x^{\frac{2}{3}} = 9. This equation involves a fractional exponent, which means that the base xx is raised to the power of 23\frac{2}{3}. To solve this equation, we need to isolate the variable xx.

Step 1: Eliminate the Fractional Exponent

To eliminate the fractional exponent, we can raise both sides of the equation to the power of 32\frac{3}{2}. This will cancel out the fractional exponent on the left-hand side.

(x^{\frac{2}{3}})^{\frac{3}{2}} = 9^{\frac{3}{2}}

Step 2: Simplify the Equation

When we raise both sides of the equation to the power of 32\frac{3}{2}, we get:

x = 9^{\frac{3}{2}}

Step 3: Evaluate the Right-Hand Side

To evaluate the right-hand side, we need to calculate the value of 9329^{\frac{3}{2}}. We can do this by first calculating the value of 9129^{\frac{1}{2}}, which is equal to 3.

9^{\frac{1}{2}} = 3

Then, we can raise 3 to the power of 3 to get:

(9^{\frac{1}{2}})^3 = 3^3 = 27

Step 4: Write the Final Solution

Therefore, the final solution to the equation x23=9x^{\frac{2}{3}} = 9 is:

x = 27

Checking for Extraneous Solutions

To check for extraneous solutions, we need to plug the solution back into the original equation and verify that it is true.

(27)^{\frac{2}{3}} = 9

When we evaluate the left-hand side, we get:

(27)^{\frac{2}{3}} = 9

Since the equation is true, we can conclude that the solution x=27x = 27 is not extraneous.

Conclusion

In this article, we solved the equation x23=9x^{\frac{2}{3}} = 9 and checked for extraneous solutions. We used the properties of exponents to eliminate the fractional exponent and isolate the variable xx. We then evaluated the right-hand side and wrote the final solution. Finally, we checked for extraneous solutions by plugging the solution back into the original equation. The solution x=27x = 27 is not extraneous, and we can conclude that it is the only solution to the equation.

Frequently Asked Questions

Q: What is a fractional exponent?

A: A fractional exponent is an exponent that is a fraction, such as 23\frac{2}{3}.

Q: How do I eliminate a fractional exponent?

A: To eliminate a fractional exponent, you can raise both sides of the equation to the power of the reciprocal of the exponent.

Q: What is an extraneous solution?

A: An extraneous solution is a solution that is not valid or is not a solution to the original equation.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you can plug the solution back into the original equation and verify that it is true.

Additional Resources

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Introduction

In our previous article, we solved the equation x23=9x^{\frac{2}{3}} = 9 and checked for extraneous solutions. In this article, we will answer some frequently asked questions about solving equations with exponents.

Q&A

Q: What is a fractional exponent?

A: A fractional exponent is an exponent that is a fraction, such as 23\frac{2}{3}. It can be written as xmnx^{\frac{m}{n}}, where mm and nn are integers and n≠0n \neq 0.

Q: How do I eliminate a fractional exponent?

A: To eliminate a fractional exponent, you can raise both sides of the equation to the power of the reciprocal of the exponent. For example, to eliminate the fractional exponent in the equation x23=9x^{\frac{2}{3}} = 9, you can raise both sides to the power of 32\frac{3}{2}.

Q: What is an extraneous solution?

A: An extraneous solution is a solution that is not valid or is not a solution to the original equation. It is a solution that is introduced during the process of solving the equation, but is not a true solution.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you can plug the solution back into the original equation and verify that it is true. If the solution is not true, then it is an extraneous solution.

Q: What is the difference between a rational exponent and a fractional exponent?

A: A rational exponent is an exponent that is a rational number, such as 23\frac{2}{3} or 2132\frac{1}{3}. A fractional exponent is a type of rational exponent where the numerator is 1. For example, 13\frac{1}{3} is a fractional exponent, but 2132\frac{1}{3} is a rational exponent.

Q: How do I simplify an expression with a rational exponent?

A: To simplify an expression with a rational exponent, you can use the properties of exponents to rewrite the expression in a simpler form. For example, to simplify the expression x23x^{\frac{2}{3}}, you can rewrite it as (x13)2(x^{\frac{1}{3}})^2.

Q: What is the order of operations for exponents?

A: The order of operations for exponents is the same as the order of operations for regular arithmetic. You should follow the order of operations as follows:

  1. Evaluate any expressions inside parentheses.
  2. Evaluate any exponential expressions.
  3. Evaluate any multiplication and division expressions from left to right.
  4. Evaluate any addition and subtraction expressions from left to right.

Q: How do I solve an equation with a rational exponent?

A: To solve an equation with a rational exponent, you can use the properties of exponents to rewrite the equation in a simpler form. For example, to solve the equation x23=9x^{\frac{2}{3}} = 9, you can raise both sides to the power of 32\frac{3}{2} to eliminate the fractional exponent.

Conclusion

In this article, we answered some frequently asked questions about solving equations with exponents. We covered topics such as fractional exponents, rational exponents, and the order of operations for exponents. We also provided examples and explanations to help illustrate the concepts.

Additional Resources

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