Simplify. Express Your Answer As The Given Base Raised To A Single Exponent. ( 4 − 6 ) − 4 \left(4^{-6}\right)^{-4} ( 4 − 6 ) − 4 □ \square □

Understanding the Problem

When dealing with exponents, it's essential to understand the rules that govern their behavior. In this problem, we're given the expression $\left(4^{-6}\right)^{-4}$ and asked to simplify it, expressing the result as a single exponent. To tackle this, we need to apply the rules of exponents, specifically the power of a power rule.

The Power of a Power Rule

The power of a power rule states that when we have an exponent raised to another exponent, we can multiply the exponents. Mathematically, this can be expressed as:

$(a^m)^n = a^{m \cdot n}$

where $a$ is the base, $m$ is the first exponent, and $n$ is the second exponent.

Applying the Power of a Power Rule

Now, let's apply this rule to the given expression $\left(4^{-6}\right)^{-4}$. According to the power of a power rule, we can multiply the exponents:

$\left(4^{-6}\right)^{-4} = 4^{-6 \cdot -4}$

Simplifying the Exponent

When we multiply two negative numbers, the result is positive. Therefore, the exponent $-6 \cdot -4$ simplifies to $24$:

$4^{-6 \cdot -4} = 4^{24}$

Conclusion

By applying the power of a power rule, we simplified the expression $\left(4^{-6}\right)^{-4}$ to $4^{24}$. This result expresses the original expression as a single exponent, as required.

Additional Examples and Practice

To further reinforce your understanding of the power of a power rule, consider the following examples:

• $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$
• $(5^{-2})^{-3} = 5^{-2 \cdot -3} = 5^6$
• $(3^4)^{-2} = 3^{4 \cdot -2} = 3^{-8}$

Practice applying the power of a power rule to different expressions to become more comfortable with this concept.

Common Mistakes to Avoid

When working with exponents, it's easy to make mistakes. Here are a few common pitfalls to watch out for:

• Failing to apply the power of a power rule when necessary
• Incorrectly multiplying or dividing exponents
• Forgetting to simplify the resulting exponent

Real-World Applications

Understanding the power of a power rule has numerous real-world applications, including:

• Calculating compound interest in finance
• Modeling population growth in biology
• Analyzing data in statistics

By mastering the power of a power rule, you'll be better equipped to tackle complex problems in various fields.

Final Thoughts

Simplifying expressions with exponents requires a solid understanding of the rules governing their behavior. By applying the power of a power rule, you can simplify complex expressions and express the result as a single exponent. Remember to practice regularly and avoid common mistakes to become proficient in this area. With time and practice, you'll become more confident in your ability to tackle even the most challenging exponent-related problems.

Frequently Asked Questions

We've covered the basics of simplifying expressions with exponents, but we know you have questions. Here are some frequently asked questions and their answers to help you better understand the power of a power rule.

Q: What is the power of a power rule?

A: The power of a power rule states that when we have an exponent raised to another exponent, we can multiply the exponents. Mathematically, this can be expressed as:

$(a^m)^n = a^{m \cdot n}$

where $a$ is the base, $m$ is the first exponent, and $n$ is the second exponent.

Q: How do I apply the power of a power rule?

A: To apply the power of a power rule, simply multiply the exponents. For example, if we have $(2^3)^4$, we can multiply the exponents as follows:

$(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$

Q: What if I have a negative exponent?

A: If you have a negative exponent, you can still apply the power of a power rule. For example, if we have $(5^{-2})^{-3}$, we can multiply the exponents as follows:

$(5^{-2})^{-3} = 5^{-2 \cdot -3} = 5^6$

Q: Can I simplify an expression with multiple exponents?

A: Yes, you can simplify an expression with multiple exponents by applying the power of a power rule multiple times. For example, if we have $(3^4)^{-2}$, we can simplify it as follows:

$(3^4)^{-2} = 3^{4 \cdot -2} = 3^{-8}$

Q: What if I have a fraction as an exponent?

A: If you have a fraction as an exponent, you can still apply the power of a power rule. For example, if we have $(2^3)^{1/2}$, we can multiply the exponents as follows:

$(2^3)^{1/2} = 2^{3 \cdot 1/2} = 2^{3/2}$

Q: Can I simplify an expression with a variable as an exponent?

A: Yes, you can simplify an expression with a variable as an exponent by applying the power of a power rule. For example, if we have $(x^2)^3$, we can simplify it as follows:

$(x^2)^3 = x^{2 \cdot 3} = x^6$

Q: What if I have a complex expression with multiple exponents?

A: If you have a complex expression with multiple exponents, you can simplify it by applying the power of a power rule multiple times. For example, if we have $(2^3)^4 \cdot (5^{-2})^{-3}$, we can simplify it as follows:

$(2^3)^4 \cdot (5^{-2})^{-3} = 2^{3 \cdot 4} \cdot 5^{-2 \cdot -3} = 2^{12} \cdot 5^6$

Q: Can I use the power of a power rule with other mathematical operations?

A: Yes, you can use the power of a power rule with other mathematical operations, such as addition and subtraction. For example, if we have $(2^3 + 5^{-2})^4$, we can simplify it as follows:

$(2^3 + 5^{-2})^4 = (8 + 1/25)^4$

However, in this case, we would need to apply the power of a power rule to the entire expression, not just the exponents.

Conclusion

We hope this Q&A article has helped you better understand the power of a power rule and how to apply it to simplify expressions with exponents. Remember to practice regularly and avoid common mistakes to become proficient in this area. With time and practice, you'll become more confident in your ability to tackle even the most challenging exponent-related problems.

Additional Resources

If you're looking for more practice problems or want to learn more about exponents, here are some additional resources:

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

Final Thoughts

Simplifying expressions with exponents requires a solid understanding of the rules governing their behavior. By applying the power of a power rule, you can simplify complex expressions and express the result as a single exponent. Remember to practice regularly and avoid common mistakes to become proficient in this area. With time and practice, you'll become more confident in your ability to tackle even the most challenging exponent-related problems.