Which Expression Is Equivalent To 4 − 6 ( 4 − 3 ) 2 \frac{4^{-6}}{\left(4^{-3}\right)^2} ( 4 − 3 ) 2 4 − 6 ​ ?A. 4 − 1 4^{-1} 4 − 1 B. 0 0 0 C. 4 2 4^2 4 2 D. 1 1 1

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Understanding Exponents and Their Properties

When dealing with exponents, it's essential to understand the properties that govern their behavior. One of the most crucial properties is the power of a power rule, which states that for any non-zero number $a$ and integers $m$ and $n$, $\left(a^m\right)^n = a^{m \cdot n}$. This rule allows us to simplify expressions involving exponents by multiplying the exponents.

Simplifying the Given Expression

The given expression is $\frac{4^{-6}}{\left(4^{-3}\right)^2}$. To simplify this expression, we can start by applying the power of a power rule to the denominator. This gives us $\frac{4^{-6}}{4^{-3 \cdot 2}} = \frac{4^{-6}}{4^{-6}}$.

Canceling Out the Common Factors

Now that we have simplified the expression, we can see that the numerator and denominator have a common factor of $4^{-6}$. We can cancel out this common factor by subtracting the exponents, which gives us $4^{-6 - (-6)} = 4^0$.

Evaluating the Final Expression

The final expression is $4^0$. According to the rules of exponents, any non-zero number raised to the power of zero is equal to 1. Therefore, $4^0 = 1$.

Conclusion

In conclusion, the expression $\frac{4^{-6}}{\left(4^{-3}\right)^2}$ is equivalent to $4^0$, which is equal to 1. This is the correct answer among the given options.

Final Answer

The final answer is D. $1$.

Understanding the Properties of Exponents

Exponents are a fundamental concept in mathematics, and understanding their properties is crucial for simplifying complex expressions. The power of a power rule is one of the most important properties of exponents, and it allows us to simplify expressions involving exponents by multiplying the exponents.

Simplifying Expressions Involving Exponents

When simplifying expressions involving exponents, it's essential to apply the power of a power rule. This rule states that for any non-zero number $a$ and integers $m$ and $n$, $\left(a^m\right)^n = a^{m \cdot n}$. By applying this rule, we can simplify complex expressions and make them easier to evaluate.

Evaluating Expressions Involving Exponents

Evaluating expressions involving exponents can be challenging, but it's essential to understand the properties of exponents to simplify these expressions. By applying the power of a power rule and canceling out common factors, we can simplify complex expressions and make them easier to evaluate.

Common Mistakes to Avoid

When simplifying expressions involving exponents, there are several common mistakes to avoid. One of the most common mistakes is not applying the power of a power rule, which can lead to incorrect simplifications. Another common mistake is not canceling out common factors, which can also lead to incorrect simplifications.

Best Practices for Simplifying Expressions Involving Exponents

To simplify expressions involving exponents, it's essential to follow best practices. One of the best practices is to apply the power of a power rule to simplify complex expressions. Another best practice is to cancel out common factors to simplify expressions.

Conclusion

In conclusion, understanding the properties of exponents is crucial for simplifying complex expressions. By applying the power of a power rule and canceling out common factors, we can simplify expressions involving exponents and make them easier to evaluate. By following best practices and avoiding common mistakes, we can simplify expressions involving exponents and arrive at the correct solution.

Final Answer

The final answer is D. $1$.

Frequently Asked Questions

Q: What is the power of a power rule in exponents?

A: The power of a power rule in exponents states that for any non-zero number $a$ and integers $m$ and $n$, $\left(a^m\right)^n = a^{m \cdot n}$. This rule allows us to simplify expressions involving exponents by multiplying the exponents.

Q: How do I simplify an expression involving exponents?

A: To simplify an expression involving exponents, you can start by applying the power of a power rule to simplify complex expressions. Then, you can cancel out common factors to simplify the expression further.

Q: What is the difference between $4^0$ and $4^{-1}$?

A: $4^0$ is equal to 1, while $4^{-1}$ is equal to $\frac{1}{4}$. This is because any non-zero number raised to the power of zero is equal to 1, while a non-zero number raised to a negative power is equal to the reciprocal of the number raised to the positive power.

Q: Can I simplify an expression involving exponents by canceling out common factors?

A: Yes, you can simplify an expression involving exponents by canceling out common factors. This is a crucial step in simplifying expressions involving exponents.

Q: What is the final answer to the expression $\frac{4^{-6}}{\left(4^{-3}\right)^2}$?

A: The final answer to the expression $\frac{4^{-6}}{\left(4^{-3}\right)^2}$ is $4^0$, which is equal to 1.

Q: How do I evaluate an expression involving exponents?

A: To evaluate an expression involving exponents, you can start by simplifying the expression using the power of a power rule and canceling out common factors. Then, you can apply the rules of exponents to evaluate the expression.

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

A: Some common mistakes to avoid when simplifying expressions involving exponents include not applying the power of a power rule, not canceling out common factors, and not following the rules of exponents.

Q: How do I follow best practices when simplifying expressions involving exponents?

A: To follow best practices when simplifying expressions involving exponents, you can start by applying the power of a power rule and canceling out common factors. Then, you can apply the rules of exponents to evaluate the expression.

Conclusion

In conclusion, simplifying expressions involving exponents requires a clear understanding of the power of a power rule and the rules of exponents. By following best practices and avoiding common mistakes, you can simplify expressions involving exponents and arrive at the correct solution.

Final Answer

The final answer is D. $1$.

Additional Resources

• Exponents and Powers
• Simplifying Expressions Involving Exponents
• Rules of Exponents

Related Questions

• What is the difference between $4^0$ and $4^{-1}$?
• Can I simplify an expression involving exponents by canceling out common factors?
• What is the final answer to the expression $\frac{4^{-6}}{\left(4^{-3}\right)^2}$?
• How do I evaluate an expression involving exponents?
• What are some common mistakes to avoid when simplifying expressions involving exponents?
• How do I follow best practices when simplifying expressions involving exponents?