Describe And Correct The Error In Simplifying The Expression.$\[ \left(2 E^{-3 X}\right)^4=\frac{1}{16 E^{12 X}} \\]Correct Simplification:The Expression \[$\left(2 E^{-3 X}\right)^4\$\] Should Be Simplified As Follows:1. Apply The

Introduction

Exponential expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will discuss the correct method for simplifying the expression $\left(2 e^{-3 x}\right)^4$ and provide a step-by-step guide on how to do it.

The Incorrect Simplification

The given expression is $\left(2 e^{-3 x}\right)^4=\frac{1}{16 e^{12 x}}$. However, this simplification is incorrect. Let's break it down and see where it went wrong.

Step 1: Apply the Power Rule

The power rule states that for any non-zero number $a$ and integers $m$ and $n$, we have $(a^m)^n = a^{m \cdot n}$. In this case, we have $\left(2 e^{-3 x}\right)^4$. Applying the power rule, we get:

$\left(2 e^{-3 x}\right)^4 = 2^4 \cdot (e^{-3 x})^4$

Step 2: Simplify the Exponential Term

Now, let's simplify the exponential term $(e^{-3 x})^4$. Using the power rule again, we get:

$(e^{-3 x})^4 = e^{-3 x \cdot 4} = e^{-12 x}$

Step 3: Simplify the Entire Expression

Now that we have simplified the exponential term, we can simplify the entire expression:

$2^4 \cdot (e^{-3 x})^4 = 2^4 \cdot e^{-12 x} = 16 \cdot e^{-12 x}$

The Correct Simplification

So, the correct simplification of the expression $\left(2 e^{-3 x}\right)^4$ is:

$\left(2 e^{-3 x}\right)^4 = 16 \cdot e^{-12 x}$

Conclusion

In conclusion, simplifying exponential expressions requires a step-by-step approach. By applying the power rule and simplifying the exponential term, we can arrive at the correct simplification. Remember, the key is to be patient and take your time when simplifying these types of expressions.

Common Mistakes to Avoid

When simplifying exponential expressions, there are several common mistakes to avoid:

• Not applying the power rule: Failing to apply the power rule can lead to incorrect simplifications.
• Not simplifying the exponential term: Failing to simplify the exponential term can lead to incorrect simplifications.
• Not being patient: Rushing through the simplification process can lead to mistakes.

Tips and Tricks

Here are some tips and tricks to help you simplify exponential expressions:

• Use the power rule: The power rule is a powerful tool for simplifying exponential expressions.
• Simplify the exponential term: Simplifying the exponential term is crucial for arriving at the correct simplification.
• Be patient: Take your time when simplifying exponential expressions, and don't rush through the process.

Practice Problems

Here are some practice problems to help you practice simplifying exponential expressions:

• $\left(3 e^{2 x}\right)^5$
• $\left(2 e^{-4 x}\right)^3$
• $\left(5 e^{x}\right)^2$

Conclusion

Introduction

In our previous article, we discussed the correct method for simplifying the expression $\left(2 e^{-3 x}\right)^4$. We also provided a step-by-step guide on how to simplify exponential expressions. In this article, we will answer some frequently asked questions (FAQs) about simplifying exponential expressions.

Q: What is the power rule for exponential expressions?

A: The power rule for exponential expressions states that for any non-zero number $a$ and integers $m$ and $n$, we have $(a^m)^n = a^{m \cdot n}$. This means that when we raise an exponential expression to a power, we can multiply the exponents.

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

A: To simplify an exponential expression with a negative exponent, we can use the rule $a^{-n} = \frac{1}{a^n}$. For example, if we have the expression $e^{-3 x}$, we can simplify it as follows:

$e^{-3 x} = \frac{1}{e^{3 x}}$

Q: What is the difference between $e^x$ and $e^{-x}$?

A: The expressions $e^x$ and $e^{-x}$ are reciprocals of each other. This means that if we have the expression $e^x$, we can simplify it as follows:

$e^x = \frac{1}{e^{-x}}$

Q: How do I simplify an exponential expression with a variable in the exponent?

A: To simplify an exponential expression with a variable in the exponent, we can use the rule $(a^m)^n = a^{m \cdot n}$. For example, if we have the expression $(e^x)^4$, we can simplify it as follows:

$(e^x)^4 = e^{x \cdot 4} = e^{4x}$

Q: What is the difference between $e^{2x}$ and $e^{-2x}$?

A: The expressions $e^{2x}$ and $e^{-2x}$ are reciprocals of each other. This means that if we have the expression $e^{2x}$, we can simplify it as follows:

$e^{2x} = \frac{1}{e^{-2x}}$

Q: How do I simplify an exponential expression with a coefficient?

A: To simplify an exponential expression with a coefficient, we can use the rule $a^m \cdot b^n = a^m \cdot b^n$. For example, if we have the expression $2e^{3x}$, we can simplify it as follows:

$2e^{3x} = 2 \cdot e^{3x}$

Q: What is the difference between $e^{3x}$ and $e^{-3x}$?

A: The expressions $e^{3x}$ and $e^{-3x}$ are reciprocals of each other. This means that if we have the expression $e^{3x}$, we can simplify it as follows:

$e^{3x} = \frac{1}{e^{-3x}}$

Conclusion

In conclusion, simplifying exponential expressions requires a step-by-step approach. By applying the power rule and simplifying the exponential term, we can arrive at the correct simplification. Remember, the key is to be patient and take your time when simplifying these types of expressions. With practice and patience, you will become proficient in simplifying exponential expressions.

Practice Problems

Here are some practice problems to help you practice simplifying exponential expressions:

• $\left(3 e^{2 x}\right)^5$
• $\left(2 e^{-4 x}\right)^3$
• $\left(5 e^{x}\right)^2$

Additional Resources

For more information on simplifying exponential expressions, check out the following resources:

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

Conclusion

In conclusion, simplifying exponential expressions is a crucial skill for students and professionals alike. By applying the power rule and simplifying the exponential term, we can arrive at the correct simplification. Remember, the key is to be patient and take your time when simplifying these types of expressions. With practice and patience, you will become proficient in simplifying exponential expressions.