Simplify The Expression: ( 4 A 2 B − 2 ) 3 \left(4 A^2 B^{-2}\right)^3 ( 4 A 2 B − 2 ) 3

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Understanding the Problem

When simplifying an expression, it's essential to apply the rules of exponents and follow the order of operations. In this case, we're given the expression $\left(4 a^2 b^{-2}\right)^3$ and need to simplify it. To start, let's break down the expression and understand its components.

Applying the Power Rule

The power rule states that for any variables $a$ and $b$ and any integers $m$ and $n$, the expression $(ab)^m = a^m b^m$ and $(a^m)^n = a^{mn}$. We can apply this rule to the given expression by raising each component to the power of 3.

Simplifying the Expression

Using the power rule, we can rewrite the expression as:

$$\left(4 a^2 b^{-2}\right)^3 = 4^3 \left(a^2\right)^3 \left(b^{-2}\right)^3$$

Evaluating the Exponents

Now, let's evaluate the exponents for each component:

• $4^3 = 64$
• $\left(a^2\right)^3 = a^6$
• $\left(b^{-2}\right)^3 = b^{-6}$

Combining the Components

We can now combine the components to simplify the expression:

$$64 a^6 b^{-6}$$

Final Simplification

To finalize the simplification, we can rewrite the expression in a more compact form by combining the variables with the same base:

$$64 a^6 b^{-6} = \frac{64 a^6}{b^6}$$

Conclusion

In this article, we simplified the expression $\left(4 a^2 b^{-2}\right)^3$ by applying the power rule and evaluating the exponents. We then combined the components to arrive at the final simplified expression: $\frac{64 a^6}{b^6}$. This demonstrates the importance of understanding and applying the rules of exponents in simplifying complex expressions.

Common Mistakes to Avoid

When simplifying expressions, it's essential to avoid common mistakes such as:

• Failing to apply the power rule correctly
• Not evaluating the exponents properly
• Not combining the components correctly

Real-World Applications

Simplifying expressions is a crucial skill in various fields, including mathematics, physics, and engineering. It's essential to understand and apply the rules of exponents to solve complex problems and arrive at accurate solutions.

Tips for Simplifying Expressions

To simplify expressions effectively, follow these tips:

• Apply the power rule correctly
• Evaluate the exponents properly
• Combine the components correctly
• Check your work for accuracy

Final Thoughts

Simplifying expressions is a fundamental concept in mathematics that requires a deep understanding of the rules of exponents. By applying the power rule and evaluating the exponents correctly, we can simplify complex expressions and arrive at accurate solutions. Remember to avoid common mistakes and follow the tips outlined in this article to simplify expressions effectively.

Additional Resources

For further learning and practice, consider the following resources:

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

Conclusion

In conclusion, simplifying the expression $\left(4 a^2 b^{-2}\right)^3$ requires a thorough understanding of the rules of exponents and the power rule. By applying these concepts and following the tips outlined in this article, we can simplify complex expressions and arrive at accurate solutions. Remember to practice and review the material to become proficient in simplifying expressions.

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Frequently Asked Questions

In this article, we'll address some common questions and concerns related to simplifying the expression $\left(4 a^2 b^{-2}\right)^3$. Whether you're a student, teacher, or simply looking for a refresher, this Q&A section is designed to provide you with the answers you need.

Q: What is the power rule, and how is it used in simplifying expressions?

A: The power rule states that for any variables $a$ and $b$ and any integers $m$ and $n$, the expression $(ab)^m = a^m b^m$ and $(a^m)^n = a^{mn}$. This rule is used to simplify expressions by raising each component to the power of the exponent.

Q: How do I apply the power rule to the given expression?

A: To apply the power rule, simply raise each component of the expression to the power of 3. In this case, we have:

$$\left(4 a^2 b^{-2}\right)^3 = 4^3 \left(a^2\right)^3 \left(b^{-2}\right)^3$$

Q: What is the difference between a positive and negative exponent?

A: A positive exponent indicates that the variable is raised to a power, while a negative exponent indicates that the variable is raised to a power and then taken to the reciprocal. In the given expression, we have a negative exponent $b^{-2}$, which means that $b$ is raised to the power of -2 and then taken to the reciprocal.

Q: How do I simplify the expression $\frac{64 a^6}{b^6}$?

A: To simplify the expression, we can rewrite it in a more compact form by combining the variables with the same base:

$$\frac{64 a^6}{b^6} = 64 a^6 b^{-6}$$

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid include:

• Failing to apply the power rule correctly
• Not evaluating the exponents properly
• Not combining the components correctly

Q: How do I check my work for accuracy?

A: To check your work for accuracy, simply re-evaluate the expression and ensure that you have applied the power rule correctly and combined the components correctly.

Q: What are some real-world applications of simplifying expressions?

A: Simplifying expressions is a crucial skill in various fields, including mathematics, physics, and engineering. It's essential to understand and apply the rules of exponents to solve complex problems and arrive at accurate solutions.

Q: What are some additional resources for learning and practicing simplifying expressions?

A: Some additional resources for learning and practicing simplifying expressions include:

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

Additional Questions and Answers

Q: Can I simplify expressions with variables that have different bases?

A: Yes, you can simplify expressions with variables that have different bases. However, you must apply the power rule correctly and combine the components correctly.

Q: How do I simplify expressions with negative exponents?

A: To simplify expressions with negative exponents, simply raise the variable to the power of the negative exponent and then take the reciprocal.

Q: Can I simplify expressions with fractions?

A: Yes, you can simplify expressions with fractions. However, you must apply the power rule correctly and combine the components correctly.

Conclusion

In this Q&A article, we've addressed some common questions and concerns related to simplifying the expression $\left(4 a^2 b^{-2}\right)^3$. Whether you're a student, teacher, or simply looking for a refresher, this article is designed to provide you with the answers you need. Remember to practice and review the material to become proficient in simplifying expressions.