Which Option Applies The Power Of A Power Rule Properly To Simplify This Expression?$\left(7^{-8}\right)^{-1}$A. $\left(7^{-8}\right)^{-4} = 7^{(-8) + (-4)} = 7^{-12} = \frac{1}{7^{12}}$B. $\left(7^{-8}\right)^{-4} = 7^{(-8) +

Introduction

The power rule is a fundamental concept in algebra that allows us to simplify expressions involving exponents. In this article, we will explore how to apply the power rule to simplify the expression $\left(7^{-8}\right)^{-1}$ and determine which option applies the power rule properly.

Understanding the Power Rule

The power rule states that for any non-zero number $a$ and integers $m$ and $n$, we have:

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

This rule allows us to simplify expressions involving exponents by multiplying the exponents.

Applying the Power Rule to the Expression

Let's apply the power rule to the expression $\left(7^{-8}\right)^{-1}$.

$$\left(7^{-8}\right)^{-1} = 7^{(-8) \cdot (-1)}$$

Using the rule for multiplying negative numbers, we have:

$$(-8) \cdot (-1) = 8$$

Therefore, we can simplify the expression as follows:

$$\left(7^{-8}\right)^{-1} = 7^8$$

Evaluating the Options

Now that we have simplified the expression, let's evaluate the options.

Option A

Option A states that $\left(7^{-8}\right)^{-4} = 7^{(-8) + (-4)} = 7^{-12} = \frac{1}{7^{12}}$.

Using the power rule, we can simplify the expression as follows:

$$\left(7^{-8}\right)^{-4} = 7^{(-8) \cdot (-4)}$$

Using the rule for multiplying negative numbers, we have:

$$(-8) \cdot (-4) = 32$$

Therefore, we can simplify the expression as follows:

$$\left(7^{-8}\right)^{-4} = 7^{32}$$

This is not equal to $7^{-12}$, so option A is incorrect.

Option B

Option B states that $\left(7^{-8}\right)^{-4} = 7^{(-8) + (-4)} = 7^{-12} = \frac{1}{7^{12}}$.

Using the power rule, we can simplify the expression as follows:

$$\left(7^{-8}\right)^{-4} = 7^{(-8) \cdot (-4)}$$

Using the rule for multiplying negative numbers, we have:

$$(-8) \cdot (-4) = 32$$

Therefore, we can simplify the expression as follows:

$$\left(7^{-8}\right)^{-4} = 7^{32}$$

This is not equal to $7^{-12}$, so option B is also incorrect.

Conclusion

In conclusion, neither option A nor option B applies the power rule properly to simplify the expression $\left(7^{-8}\right)^{-1}$. The correct simplification is $7^8$.

Tips and Tricks

When applying the power rule, make sure to multiply the exponents correctly. Also, be careful when multiplying negative numbers, as the rule for multiplying negative numbers is different from the rule for multiplying positive numbers.

Common Mistakes

One common mistake when applying the power rule is to forget to multiply the exponents. Another common mistake is to multiply the exponents incorrectly, such as multiplying two negative numbers as if they were positive numbers.

Real-World Applications

The power rule has many real-world applications, such as simplifying expressions in physics and engineering. For example, when calculating the force of a spring, we may need to simplify expressions involving exponents.

Practice Problems

Here are some practice problems to help you apply the power rule:

1. Simplify the expression $\left(2^3\right)^4$.
2. Simplify the expression $\left(3^{-2}\right)^{-3}$.
3. Simplify the expression $\left(4^2\right)^{-1}$.

Answer Key

1. $\left(2^3\right)^4 = 2^{3 \cdot 4} = 2^{12}$
2. $\left(3^{-2}\right)^{-3} = 3^{(-2) \cdot (-3)} = 3^6$
3. $\left(4^2\right)^{-1} = 4^{2 \cdot (-1)} = 4^{-2} = \frac{1}{4^2}$
   Frequently Asked Questions: Simplifying Expressions with the Power Rule ====================================================================

Q: What is the power rule in algebra?

A: The power rule is a fundamental concept in algebra that allows us to simplify expressions involving exponents. It states that for any non-zero number $a$ and integers $m$ and $n$, we have:

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

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

A: To apply the power rule, simply multiply the exponents. For example, if we have the expression $\left(2^3\right)^4$, we can simplify it as follows:

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

Q: What if I have a negative exponent? How do I apply the power rule?

A: If you have a negative exponent, you can still apply the power rule. For example, if we have the expression $\left(3^{-2}\right)^{-3}$, we can simplify it as follows:

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

Q: What if I have a fraction as an exponent? How do I apply the power rule?

A: If you have a fraction as an exponent, you can still apply the power rule. For example, if we have the expression $\left(4^{\frac{1}{2}}\right)^{-2}$, we can simplify it as follows:

$$\left(4^{\frac{1}{2}}\right)^{-2} = 4^{\frac{1}{2} \cdot (-2)} = 4^{-1} = \frac{1}{4}$$

Q: Can I apply the power rule to expressions with more than one exponent?

A: Yes, you can apply the power rule to expressions with more than one exponent. For example, if we have the expression $\left(2^3 \cdot 3^2\right)^4$, we can simplify it as follows:

$$\left(2^3 \cdot 3^2\right)^4 = \left(2^3\right)^4 \cdot \left(3^2\right)^4 = 2^{3 \cdot 4} \cdot 3^{2 \cdot 4} = 2^{12} \cdot 3^8$$

Q: What are some common mistakes to avoid when applying the power rule?

A: Some common mistakes to avoid when applying the power rule include:

• Forgetting to multiply the exponents
• Multiplying the exponents incorrectly
• Not considering the sign of the exponents

Q: How do I know when to use the power rule?

A: You should use the power rule whenever you have an expression with exponents that you want to simplify. The power rule is a powerful tool for simplifying expressions and can be used in a wide range of mathematical applications.

Q: Can I use the power rule to simplify expressions with variables?

A: Yes, you can use the power rule to simplify expressions with variables. For example, if we have the expression $\left(x^2\right)^3$, we can simplify it as follows:

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

Q: What are some real-world applications of the power rule?

A: The power rule has many real-world applications, such as simplifying expressions in physics and engineering. For example, when calculating the force of a spring, we may need to simplify expressions involving exponents.

Q: Can I use the power rule to simplify expressions with fractions?

A: Yes, you can use the power rule to simplify expressions with fractions. For example, if we have the expression $\left(\frac{1}{2}\right)^4$, we can simplify it as follows:

$$\left(\frac{1}{2}\right)^4 = \frac{1}{2^4} = \frac{1}{16}$$

Q: What are some tips for mastering the power rule?

A: Some tips for mastering the power rule include:

• Practicing, practicing, practicing: The more you practice applying the power rule, the more comfortable you will become with it.
• Understanding the rules: Make sure you understand the rules for multiplying exponents and fractions.
• Using visual aids: Visual aids such as diagrams and charts can help you understand the power rule and how to apply it.
• Breaking down complex expressions: Break down complex expressions into simpler ones to make them easier to work with.