Solve For { X $} : : : { 6^{-3x-2} = 36 \}

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Introduction

Solving exponential equations can be a challenging task, but with the right approach, it can be made easier. In this article, we will focus on solving the equation $6^{-3x-2} = 36$. This equation involves a negative exponent and a base of 6, which can be simplified using the properties of exponents.

Understanding the Equation

The given equation is $6^{-3x-2} = 36$. To solve for $x$, we need to isolate the variable $x$ on one side of the equation. The first step is to simplify the equation by expressing 36 as a power of 6.

Simplifying the Equation

We know that $36 = 6^2$. Therefore, we can rewrite the equation as $6^{-3x-2} = 6^2$. Now, we can equate the exponents on both sides of the equation.

Equating Exponents

Since the bases are the same (6), we can equate the exponents: $-3x-2 = 2$. Now, we need to solve for $x$.

Solving for $x$

To solve for $x$, we need to isolate the variable $x$ on one side of the equation. We can start by adding 2 to both sides of the equation: $-3x-2+2 = 2+2$. This simplifies to $-3x = 4$.

Isolating $x$

Next, we need to isolate $x$ by dividing both sides of the equation by -3: $\frac{-3x}{-3} = \frac{4}{-3}$. This simplifies to $x = -\frac{4}{3}$.

Conclusion

In this article, we solved the equation $6^{-3x-2} = 36$ by simplifying the equation, equating exponents, and isolating the variable $x$. The final solution is $x = -\frac{4}{3}$.

Properties of Exponents

Exponents are a fundamental concept in mathematics, and understanding their properties is essential for solving equations involving exponents. Here are some key properties of exponents:

• Product of Powers: When multiplying two powers with the same base, we can add the exponents: $a^m \cdot a^n = a^{m+n}$.
• Power of a Power: When raising a power to another power, we can multiply the exponents: $(a^m)^n = a^{m \cdot n}$.
• Negative Exponent: A negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base: $a^{-m} = \frac{1}{a^m}$.

Examples of Solving Exponential Equations

Here are some examples of solving exponential equations:

Example 1

Solve the equation $2^x = 8$.

Step 1: Simplify the Equation

We know that $8 = 2^3$. Therefore, we can rewrite the equation as $2^x = 2^3$.

Step 2: Equate Exponents

Since the bases are the same (2), we can equate the exponents: $x = 3$.

Step 3: Conclusion

The final solution is $x = 3$.

Example 2

Solve the equation $3^{-2x} = \frac{1}{9}$.

Step 1: Simplify the Equation

We know that $\frac{1}{9} = 3^{-2}$. Therefore, we can rewrite the equation as $3^{-2x} = 3^{-2}$.

Step 2: Equate Exponents

Since the bases are the same (3), we can equate the exponents: $-2x = -2$.

Step 3: Solve for $x$

To solve for $x$, we need to isolate the variable $x$ on one side of the equation. We can start by dividing both sides of the equation by -2: $\frac{-2x}{-2} = \frac{-2}{-2}$. This simplifies to $x = 1$.

Step 4: Conclusion

The final solution is $x = 1$.

Tips for Solving Exponential Equations

Here are some tips for solving exponential equations:

• Simplify the Equation: Before solving the equation, simplify it by expressing the bases as powers of a common base.
• Equate Exponents: Since the bases are the same, equate the exponents on both sides of the equation.
• Isolate the Variable: Isolate the variable on one side of the equation by adding, subtracting, multiplying, or dividing both sides of the equation by a constant.
• Check the Solution: Check the solution by plugging it back into the original equation.

Conclusion

Solving exponential equations can be a challenging task, but with the right approach, it can be made easier. By simplifying the equation, equating exponents, and isolating the variable, we can solve exponential equations. Remember to check the solution by plugging it back into the original equation. With practice and patience, you can become proficient in solving exponential equations.

Frequently Asked Questions

Exponential equations can be a challenging topic, but with the right guidance, you can become proficient in solving them. Here are some frequently asked questions about exponential equations:

Q: What is an exponential equation?

A: An exponential equation is an equation that involves a variable in the exponent of a number. For example, $2^x = 8$ is an exponential equation.

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, you can express the bases as powers of a common base. For example, $8 = 2^3$, so we can rewrite the equation $2^x = 8$ as $2^x = 2^3$.

Q: How do I equate exponents?

A: Since the bases are the same, you can equate the exponents on both sides of the equation. For example, in the equation $2^x = 2^3$, we can equate the exponents: $x = 3$.

Q: How do I isolate the variable?

A: To isolate the variable, you can add, subtract, multiply, or divide both sides of the equation by a constant. For example, in the equation $-3x = 4$, we can isolate the variable by dividing both sides of the equation by -3: $x = -\frac{4}{3}$.

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

A: Some common mistakes to avoid when solving exponential equations include:

• Not simplifying the equation: Make sure to simplify the equation by expressing the bases as powers of a common base.
• Not equating exponents: Since the bases are the same, make sure to equate the exponents on both sides of the equation.
• Not isolating the variable: Make sure to isolate the variable on one side of the equation by adding, subtracting, multiplying, or dividing both sides of the equation by a constant.
• Not checking the solution: Make sure to check the solution by plugging it back into the original equation.

Q: What are some tips for solving exponential equations?

A: Some tips for solving exponential equations include:

• Simplify the equation: Before solving the equation, simplify it by expressing the bases as powers of a common base.
• Equate exponents: Since the bases are the same, equate the exponents on both sides of the equation.
• Isolate the variable: Isolate the variable on one side of the equation by adding, subtracting, multiplying, or dividing both sides of the equation by a constant.
• Check the solution: Check the solution by plugging it back into the original equation.

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

A: Exponential equations have many real-world applications, including:

• Population growth: Exponential equations can be used to model population growth, where the population grows at a rate proportional to the current population.
• Financial modeling: Exponential equations can be used to model financial growth, where the value of an investment grows at a rate proportional to the current value.
• Science and engineering: Exponential equations can be used to model many scientific and engineering phenomena, including chemical reactions, electrical circuits, and population dynamics.

Q: How can I practice solving exponential equations?

A: There are many ways to practice solving exponential equations, including:

• Solving problems: Practice solving exponential equations by working through problems in a textbook or online resource.
• Using online resources: Use online resources, such as Khan Academy or Wolfram Alpha, to practice solving exponential equations.
• Working with a tutor: Work with a tutor or teacher to practice solving exponential equations and get feedback on your work.

Conclusion

Exponential equations can be a challenging topic, but with the right guidance and practice, you can become proficient in solving them. Remember to simplify the equation, equate exponents, and isolate the variable to solve exponential equations. With practice and patience, you can become proficient in solving exponential equations and apply them to real-world problems.