Use The Power Rule Of Exponents To Simplify The Expression $\left(6^2\right)^4$. $\square$

Understanding the Power Rule of Exponents

The power rule of exponents is a fundamental concept in algebra that allows us to simplify complex exponential expressions. It states that when we have an expression in the form of $(a^m)^n$, we can simplify it to $a^{m \cdot n}$. This rule is essential in simplifying expressions with multiple exponents and is widely used in various mathematical applications.

Applying the Power Rule to the Given Expression

In this problem, we are given the expression $\left(6^2\right)^4$. To simplify this expression using the power rule, we need to apply the rule to the given expression. The power rule states that we can multiply the exponents when the base is the same. In this case, the base is $6$ and the exponents are $2$ and $4$.

Simplifying the Expression

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

$\left(6^2\right)^4 = 6^{2 \cdot 4} = 6^8$

This means that the expression $\left(6^2\right)^4$ can be simplified to $6^8$. This is a much simpler expression than the original one, and it is easier to work with.

Why the Power Rule Works

The power rule works because of the way exponents are defined. When we have an expression in the form of $a^m$, we can think of it as multiplying the base $a$ by itself $m$ times. For example, $6^2$ can be thought of as $6 \cdot 6$. When we raise this expression to the power of $n$, we are essentially multiplying the result by itself $n$ times.

Real-World Applications of the Power Rule

The power rule has many real-world applications in various fields, including science, engineering, and economics. For example, in physics, the power rule is used to calculate the energy of a system, while in engineering, it is used to design and optimize systems. In economics, the power rule is used to model population growth and other economic phenomena.

Common Mistakes to Avoid

When applying the power rule, there are several common mistakes to avoid. One of the most common mistakes is to forget to multiply the exponents. This can lead to incorrect results and can be frustrating to debug. Another common mistake is to apply the power rule to expressions that are not in the correct form. For example, the power rule cannot be applied to expressions that have different bases.

Conclusion

In conclusion, the power rule of exponents is a powerful tool for simplifying complex exponential expressions. By applying the power rule, we can simplify expressions like $\left(6^2\right)^4$ to $6^8$. This rule is essential in various mathematical applications and has many real-world applications. By understanding the power rule and how to apply it correctly, we can simplify complex expressions and solve problems more efficiently.

Examples and Practice Problems

Here are some examples and practice problems to help you understand the power rule better:

Example 1

Simplify the expression $\left(3^4\right)^2$ using the power rule.

Solution

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

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

Example 2

Simplify the expression $\left(2^3\right)^5$ using the power rule.

Solution

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

$\left(2^3\right)^5 = 2^{3 \cdot 5} = 2^{15}$

Practice Problems

1. Simplify the expression $\left(4^2\right)^3$ using the power rule.
2. Simplify the expression $\left(5^4\right)^2$ using the power rule.
3. Simplify the expression $\left(2^5\right)^4$ using the power rule.

Answer Key

1. $4^{2 \cdot 3} = 4^6$
2. $5^{4 \cdot 2} = 5^8$
3. $2^{5 \cdot 4} = 2^{20}$
   

Q: What is the power rule of exponents?

A: The power rule of exponents is a fundamental concept in algebra that allows us to simplify complex exponential expressions. It states that when we have an expression in the form of $(a^m)^n$, we can simplify it to $a^{m \cdot n}$.

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

A: To apply the power rule, you need to multiply the exponents when the base is the same. For example, if you have the expression $\left(6^2\right)^4$, you can simplify it to $6^{2 \cdot 4} = 6^8$.

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 and applying the power rule to expressions that are not in the correct form. For example, the power rule cannot be applied to expressions that have different bases.

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

A: Yes, you can apply the power rule to expressions with negative exponents. For example, if you have the expression $\left(2^{-3}\right)^4$, you can simplify it to $2^{-3 \cdot 4} = 2^{-12}$.

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 you have the expression $\left(3^{1/2}\right)^4$, you can simplify it to $3^{1/2 \cdot 4} = 3^2$.

Q: How do I simplify expressions with multiple exponents?

A: To simplify expressions with multiple exponents, you need to apply the power rule multiple times. For example, if you have the expression $\left(6^2\right)^4 \cdot \left(6^3\right)^2$, you can simplify it to $6^{2 \cdot 4} \cdot 6^{3 \cdot 2} = 6^8 \cdot 6^6 = 6^{8 + 6} = 6^{14}$.

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

A: Yes, you can apply the power rule to expressions with variables as exponents. For example, if you have the expression $\left(x^2\right)^4$, you can simplify it to $x^{2 \cdot 4} = x^8$.

Q: How do I simplify expressions with exponents and fractions?

A: To simplify expressions with exponents and fractions, you need to apply the power rule and simplify the fraction. For example, if you have the expression $\left(\frac{2}{3}\right)^4$, you can simplify it to $\frac{2^4}{3^4} = \frac{16}{81}$.

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

A: Yes, you can apply the power rule to expressions with radicals. For example, if you have the expression $\left(\sqrt{2}\right)^4$, you can simplify it to $\left(2^{1/2}\right)^4 = 2^{1/2 \cdot 4} = 2^2 = 4$.

Q: How do I simplify expressions with exponents and logarithms?

A: To simplify expressions with exponents and logarithms, you need to apply the power rule and the properties of logarithms. For example, if you have the expression $\log_{10} \left(6^8\right)$, you can simplify it to $8 \log_{10} 6$.

Q: Can I apply the power rule to expressions with complex numbers?

A: Yes, you can apply the power rule to expressions with complex numbers. For example, if you have the expression $\left(2 + 3i\right)^4$, you can simplify it to $\left(2 + 3i\right)^{4} = \left(2 + 3i\right)^{2 \cdot 2} = \left(\left(2 + 3i\right)^2\right)^2 = \left(4 + 12i + 9i^2\right)^2 = \left(4 + 12i - 9\right)^2 = \left(-5 + 12i\right)^2 = 25 - 120i + 144i^2 = 25 - 120i - 144 = -119 - 120i$.

Conclusion

In conclusion, the power rule of exponents is a powerful tool for simplifying complex exponential expressions. By understanding the power rule and how to apply it correctly, you can simplify expressions like $\left(6^2\right)^4$ to $6^8$. This rule is essential in various mathematical applications and has many real-world applications. By understanding the power rule and how to apply it correctly, you can simplify complex expressions and solve problems more efficiently.