Solve The Equation: $\[ 6^{-2a} = 6^{2-3a} \\]

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Introduction

When it comes to solving 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 focus on solving the equation 6^(-2a) = 6^(2-3a) using algebraic techniques.

Understanding Exponents

Before we dive into solving the equation, let's review the basics of exponents. An exponent is a small number that is raised to the power of a larger number. For example, 6^2 means 6 squared, or 6 multiplied by itself 2 times. The exponent can be positive, negative, or even a fraction.

Properties of Exponents

There are several properties of exponents that we need to know when solving equations involving exponents. These properties include:

  • Product of Powers: When multiplying two numbers with the same base, we add the exponents. For example, 6^2 * 6^3 = 6^(2+3) = 6^5.
  • Power of a Power: When raising a number with an exponent to another power, we multiply the exponents. For example, (62)3 = 6^(2*3) = 6^6.
  • Quotient of Powers: When dividing two numbers with the same base, we subtract the exponents. For example, 6^3 / 6^2 = 6^(3-2) = 6^1.

Solving the Equation

Now that we have reviewed the properties of exponents, let's focus on solving the equation 6^(-2a) = 6^(2-3a). To solve this equation, we need to use the property of exponents that states that if two numbers with the same base are equal, then their exponents must be equal.

Setting Up the Equation

We can set up the equation by equating the exponents:

-2a = 2-3a

Isolating the Variable

To isolate the variable, we need to get all the terms with the variable on one side of the equation. We can do this by adding 3a to both sides of the equation:

-2a + 3a = 2

Simplifying the Equation

Now that we have isolated the variable, we can simplify the equation by combining like terms:

a = 2

Conclusion

In this article, we have solved the equation 6^(-2a) = 6^(2-3a) using algebraic techniques. We have reviewed the properties of exponents and used them to set up and solve the equation. The final solution is a = 2.

Additional Tips and Tricks

When solving equations involving exponents, it's essential to remember the following tips and tricks:

  • Use the properties of exponents: When solving equations involving exponents, it's essential to use the properties of exponents to simplify the equation.
  • Isolate the variable: To solve the equation, we need to isolate the variable by getting all the terms with the variable on one side of the equation.
  • Simplify the equation: Once we have isolated the variable, we can simplify the equation by combining like terms.

Frequently Asked Questions

  • What is the property of exponents that states that if two numbers with the same base are equal, then their exponents must be equal? The property of exponents that states that if two numbers with the same base are equal, then their exponents must be equal is called the one-to-one property.
  • How do I simplify the equation once I have isolated the variable? To simplify the equation once you have isolated the variable, you can combine like terms by adding or subtracting the coefficients of the variable.

Final Thoughts

Solving equations involving exponents can be challenging, but with practice and patience, you can master the techniques and become proficient in solving these types of equations. Remember to use the properties of exponents, isolate the variable, and simplify the equation to get the final solution.

Introduction

In our previous article, we solved the equation 6^(-2a) = 6^(2-3a) using algebraic techniques. In this article, we will provide a Q&A section to help you better understand the solution and provide additional tips and tricks for solving equations involving exponents.

Q&A

Q: What is the property of exponents that states that if two numbers with the same base are equal, then their exponents must be equal?

A: The property of exponents that states that if two numbers with the same base are equal, then their exponents must be equal is called the one-to-one property.

Q: How do I simplify the equation once I have isolated the variable?

A: To simplify the equation once you have isolated the variable, you can combine like terms by adding or subtracting the coefficients of the variable.

Q: What if the equation has multiple variables?

A: If the equation has multiple variables, you can use the same techniques as before to isolate each variable. However, you may need to use additional steps to solve the equation.

Q: Can I use the properties of exponents to solve equations with different bases?

A: Yes, you can use the properties of exponents to solve equations with different bases. However, you will need to use the change of base formula to convert the bases to a common base.

Q: How do I know which property of exponents to use?

A: To determine which property of exponents to use, you need to analyze the equation and identify the base and the exponents. Then, you can use the properties of exponents to simplify the equation.

Q: What if the equation has a negative exponent?

A: If the equation has a negative exponent, you can use the negative exponent rule to rewrite the equation with a positive exponent.

Q: Can I use the properties of exponents to solve equations with fractions?

A: Yes, you can use the properties of exponents to solve equations with fractions. However, you will need to use the fractional exponent rule to simplify the equation.

Additional Tips and Tricks

  • Use the properties of exponents: When solving equations involving exponents, it's essential to use the properties of exponents to simplify the equation.
  • Isolate the variable: To solve the equation, we need to isolate the variable by getting all the terms with the variable on one side of the equation.
  • Simplify the equation: Once we have isolated the variable, we can simplify the equation by combining like terms.
  • Use the change of base formula: If the equation has different bases, you can use the change of base formula to convert the bases to a common base.
  • Use the negative exponent rule: If the equation has a negative exponent, you can use the negative exponent rule to rewrite the equation with a positive exponent.
  • Use the fractional exponent rule: If the equation has fractions, you can use the fractional exponent rule to simplify the equation.

Conclusion

Solving equations involving exponents can be challenging, but with practice and patience, you can master the techniques and become proficient in solving these types of equations. Remember to use the properties of exponents, isolate the variable, and simplify the equation to get the final solution.

Final Thoughts

We hope this Q&A article has helped you better understand the solution to the equation 6^(-2a) = 6^(2-3a) and provided additional tips and tricks for solving equations involving exponents. If you have any further questions or need additional help, please don't hesitate to ask.

Related Articles

  • Solving Equations with Exponents: This article provides a comprehensive guide to solving equations with exponents.
  • Properties of Exponents: This article reviews the properties of exponents and how to use them to simplify equations.
  • Algebraic Techniques: This article provides a review of algebraic techniques for solving equations, including isolating the variable and simplifying the equation.

Resources

  • Mathway: This online math problem solver can help you solve equations involving exponents.
  • Khan Academy: This online resource provides video lessons and practice exercises for solving equations involving exponents.
  • Math Open Reference: This online reference book provides a comprehensive guide to solving equations involving exponents.