Solve The Equation:$ 9^{-3x} \cdot 9^x = 27 $

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Introduction

In mathematics, solving equations is a fundamental concept that involves manipulating algebraic expressions to isolate the variable. In this article, we will focus on solving the equation $9^{-3x} \cdot 9^x = 27$. This equation involves exponentiation and multiplication, and we will use various mathematical techniques to simplify and solve it.

Understanding the Equation

The given equation is $9^{-3x} \cdot 9^x = 27$. To begin solving this equation, we need to understand the properties of exponents and how they interact with multiplication. The equation involves two terms with exponents: $9^{-3x}$ and $9^x$. We can use the property of exponents that states $a^m \cdot a^n = a^{m+n}$ to simplify the left-hand side of the equation.

Simplifying the Equation

Using the property of exponents, we can rewrite the left-hand side of the equation as:

$$9^{-3x} \cdot 9^x = 9^{-3x + x} = 9^{-2x}$$

Now, the equation becomes $9^{-2x} = 27$. We can further simplify this equation by expressing $27$ as a power of $9$. Since $27 = 9^{\frac{3}{2}}$, we can rewrite the equation as:

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

Equating Exponents

Since the bases of both sides of the equation are the same ($9$), we can equate the exponents:

$$-2x = \frac{3}{2}$$

Solving for x

To solve for $x$, we can divide both sides of the equation by $-2$:

$$x = -\frac{3}{4}$$

Conclusion

In this article, we solved the equation $9^{-3x} \cdot 9^x = 27$ using various mathematical techniques. We simplified the equation by using the property of exponents, equated the exponents, and solved for $x$. The solution to the equation is $x = -\frac{3}{4}$.

Additional Tips and Tricks

• When solving equations involving exponents, it's essential to use the properties of exponents to simplify the expressions.
• Equating the exponents of both sides of the equation is a crucial step in solving equations involving exponents.
• When solving for $x$, make sure to isolate the variable by performing the necessary operations.

Real-World Applications

Solving equations involving exponents has numerous real-world applications in various fields, including:

• Physics: Exponents are used to describe the behavior of physical systems, such as the motion of objects under the influence of gravity.
• Engineering: Exponents are used to describe the behavior of electrical circuits and mechanical systems.
• Computer Science: Exponents are used in algorithms and data structures to solve complex problems.

Final Thoughts

Solving the equation $9^{-3x} \cdot 9^x = 27$ requires a deep understanding of the properties of exponents and how they interact with multiplication. By using various mathematical techniques, we can simplify and solve the equation to find the value of $x$. This article provides a step-by-step guide to solving the equation and highlights the importance of understanding the properties of exponents in mathematics.

Frequently Asked Questions

• Q: What is the property of exponents that states $a^m \cdot a^n = a^{m+n}$? A: This property states that when multiplying two terms with the same base, we can add the exponents.
• Q: How do we simplify the left-hand side of the equation $9^{-3x} \cdot 9^x = 27$? A: We can use the property of exponents to rewrite the left-hand side as $9^{-3x + x} = 9^{-2x}$.
• Q: What is the solution to the equation $9^{-3x} \cdot 9^x = 27$? A: The solution to the equation is $x = -\frac{3}{4}$.
  

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Introduction

In our previous article, we solved the equation $9^{-3x} \cdot 9^x = 27$ using various mathematical techniques. In this article, we will provide a Q&A section to address some of the common questions and concerns that readers may have.

Q&A

Q: What is the property of exponents that states $a^m \cdot a^n = a^{m+n}$?

A: This property states that when multiplying two terms with the same base, we can add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.

Q: How do we simplify the left-hand side of the equation $9^{-3x} \cdot 9^x = 27$?

A: We can use the property of exponents to rewrite the left-hand side as $9^{-3x + x} = 9^{-2x}$.

Q: What is the solution to the equation $9^{-3x} \cdot 9^x = 27$?

A: The solution to the equation is $x = -\frac{3}{4}$.

Q: Can you explain the concept of exponents in more detail?

A: Exponents are a way of representing repeated multiplication. For example, $a^m$ means $a$ multiplied by itself $m$ times. Exponents can be positive, negative, or zero.

Q: How do we handle negative exponents?

A: Negative exponents can be handled by taking the reciprocal of the base. For example, $a^{-m} = \frac{1}{a^m}$.

Q: Can you provide more examples of solving equations involving exponents?

A: Yes, here are a few more examples:

• $2^{-3x} \cdot 2^x = 8$
• $5^{2x} \cdot 5^{-x} = 25$
• $3^{-x} \cdot 3^x = 9$

Q: What are some real-world applications of solving equations involving exponents?

A: Solving equations involving exponents has numerous real-world applications in various fields, including:

• Physics: Exponents are used to describe the behavior of physical systems, such as the motion of objects under the influence of gravity.
• Engineering: Exponents are used to describe the behavior of electrical circuits and mechanical systems.
• Computer Science: Exponents are used in algorithms and data structures to solve complex problems.

Q: Can you provide more tips and tricks for solving equations involving exponents?

A: Yes, here are a few more tips and tricks:

• Use the property of exponents to simplify the left-hand side of the equation.
• Equate the exponents of both sides of the equation.
• Use the reciprocal of the base to handle negative exponents.
• Practice, practice, practice!

Conclusion

In this article, we provided a Q&A section to address some of the common questions and concerns that readers may have. We also provided more examples of solving equations involving exponents and highlighted the importance of understanding the properties of exponents in mathematics. By following the tips and tricks provided in this article, readers can improve their skills in solving equations involving exponents.

Additional Resources

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram Alpha: Exponents and Exponential Functions

Final Thoughts

Solving equations involving exponents requires a deep understanding of the properties of exponents and how they interact with multiplication. By using various mathematical techniques, we can simplify and solve the equation to find the value of $x$. This article provides a step-by-step guide to solving the equation and highlights the importance of understanding the properties of exponents in mathematics.