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

Understanding the Power Rule

The power rule is a fundamental concept in algebra that allows us to simplify expressions involving exponents. It states that when we raise a power to a power, we multiply the exponents. In other words, if we have an expression of the form $(a^m)^n$, we can simplify it to $a^{mn}$. This rule is essential in simplifying complex expressions and is a crucial tool in algebra.

Applying the Power Rule to the Given Expression

The given expression is $\left(7^{-8}\right)^{-4}$. To simplify this expression, we need to apply the power rule. We can start by raising the exponent $-8$ to the power of $-4$. This gives us $7^{(-8)+(-4)}$. Now, we can simplify the expression by multiplying the exponents.

Simplifying the Expression

When we multiply the exponents, we get $7^{-12}$. This is because $-8 \times -4 = 32$, but since we are multiplying two negative numbers, the result is positive. Therefore, $7^{-12}$ is equivalent to $\frac{1}{7^{12}}$.

Evaluating the Options

Now that we have simplified the expression, we can evaluate the options.

Option A

Option A states that $\left(7^{-8}\right)^{-4}=7^{(-8)+(-4)}=7^{-12}=\frac{1}{7^{12}}$. This option is correct because we have applied the power rule correctly and simplified the expression to $\frac{1}{7^{12}}$.

Option B

Option B is incorrect because it does not apply the power rule correctly. It states that $\left(7^{-8}\right)^{-4}=7^{(-8)+(-4)}=7^{-32}$. This is incorrect because we multiplied the exponents incorrectly.

Conclusion

In conclusion, the correct option is A. $\left(7^{-8}\right)^{-4}=7^{(-8)+(-4)}=7^{-12}=\frac{1}{7^{12}}$. This option applies the power rule correctly and simplifies the expression to $\frac{1}{7^{12}}$.

Understanding the Power Rule in Different Scenarios

The power rule is a versatile concept that can be applied in different scenarios. Here are a few examples:

Example 1

Suppose we have the expression $(2^3)^4$. To simplify this expression, we can apply the power rule by raising the exponent $3$ to the power of $4$. This gives us $2^{3 \times 4} = 2^{12}$.

Example 2

Suppose we have the expression $(x^2)^3$. To simplify this expression, we can apply the power rule by raising the exponent $2$ to the power of $3$. This gives us $x^{2 \times 3} = x^6$.

Example 3

Suppose we have the expression $(a^{-2})^3$. To simplify this expression, we can apply the power rule by raising the exponent $-2$ to the power of $3$. This gives us $a^{-2 \times 3} = a^{-6}$.

Common Mistakes to Avoid

When applying the power rule, there are a few common mistakes to avoid:

Mistake 1

Not applying the power rule correctly. This can lead to incorrect simplifications.

Mistake 2

Multiplying the exponents incorrectly. This can lead to incorrect simplifications.

Mistake 3

Not considering the sign of the exponents. This can lead to incorrect simplifications.

Tips for Applying the Power Rule

Here are a few tips for applying the power rule:

Tip 1

Make sure to apply the power rule correctly by raising the exponent to the power of the other exponent.

Tip 2

Make sure to multiply the exponents correctly.

Tip 3

Make sure to consider the sign of the exponents.

Conclusion

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 when we raise a power to a power, we multiply the exponents.

Q: How do I apply the power rule?

A: To apply the power rule, you need to raise the exponent to the power of the other exponent. For example, if we have the expression $(a^m)^n$, we can simplify it to $a^{mn}$.

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:

• Not applying the power rule correctly
• Multiplying the exponents incorrectly
• Not considering the sign of the exponents

Q: How do I simplify expressions with negative exponents?

A: To simplify expressions with negative exponents, you need to apply the power rule and then simplify the resulting expression. For example, if we have the expression $(a^{-m})^n$, we can simplify it to $a^{-mn}$.

Q: Can I apply the power rule to expressions with fractional exponents?

A: Yes, you can apply the power rule to expressions with fractional exponents. For example, if we have the expression $(a^{m/n})^p$, we can simplify it to $a^{(m/n)p}$.

Q: How do I apply the power rule to expressions with multiple exponents?

A: To apply the power rule to expressions with multiple exponents, you need to apply the power rule to each exponent separately and then simplify the resulting expression. For example, if we have the expression $(a^m)^n(b^p)^q$, we can simplify it to $a^{mn}b^{pq}$.

Q: Can I apply the power rule to expressions with variables in the exponent?

A: Yes, you can apply the power rule to expressions with variables in the exponent. For example, if we have the expression $(x^m)^n$, we can simplify it to $x^{mn}$.

Q: How do I check my work when applying the power rule?

A: To check your work when applying the power rule, you need to simplify the expression and then check if it is equivalent to the original expression. For example, if we have the expression $(a^m)^n$, we can simplify it to $a^{mn}$ and then check if it is equivalent to the original expression.

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

A: The power rule has many real-world applications, including:

• Simplifying complex expressions in algebra and calculus
• Solving equations and inequalities in algebra and calculus
• Modeling real-world phenomena in physics and engineering
• Optimizing functions in economics and finance

Conclusion

In conclusion, the power rule is a fundamental concept in algebra that allows us to simplify expressions involving exponents. By applying the power rule correctly, we can simplify complex expressions and arrive at the correct solution. Remember to avoid common mistakes and follow the tips for applying the power rule. With practice and patience, you will become proficient in applying the power rule and simplifying expressions.