Simplify The Expression $\left(6 2\right) 4$.Enter The Correct Answer In The Box.

Introduction

In mathematics, simplifying expressions is an essential skill that helps us solve problems efficiently. When dealing with exponents, it's crucial to understand the rules of exponentiation to simplify complex expressions. In this article, we will focus on simplifying the expression $\left(6^2\right)4$ using the rules of exponentiation.

Understanding Exponents

Before we dive into simplifying the expression, let's review the basics of exponents. An exponent is a small number that is written above and to the right of a larger number, indicating how many times the larger number should be multiplied by itself. For example, in the expression $a^b$, the exponent $b$ indicates that the base $a$ should be multiplied by itself $b$ times.

The Power of a Power Rule

One of the most important rules of exponentiation is the power of a power rule, which states that when we raise a power to another power, we multiply the exponents. This rule can be expressed as:

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

where $a$ is the base, $m$ and $n$ are the exponents, and the result is a new exponent.

Applying the Power of a Power Rule

Now that we have reviewed the power of a power rule, let's apply it to the expression $\left(6^2\right)4$. Using the rule, we can simplify the expression as follows:

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

Simplifying the Expression

Now that we have simplified the expression using the power of a power rule, let's calculate the final result. To do this, we need to multiply the base $6$ by itself $8$ times:

$$6^8 = 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 = 1679616$$

Conclusion

In this article, we simplified the expression $\left(6^2\right)4$ using the power of a power rule. We reviewed the basics of exponents and applied the rule to simplify the expression. Finally, we calculated the final result by multiplying the base $6$ by itself $8$ times. By following these steps, we can simplify complex expressions and solve problems efficiently.

Frequently Asked Questions

• What is the power of a power rule? The power of a power rule states that when we raise a power to another power, we multiply the exponents.
• How do we apply the power of a power rule? To apply the power of a power rule, we multiply the exponents of the two powers.
• What is the final result of the expression $\left(6^2\right)4$? The final result of the expression $\left(6^2\right)4$ is $1679616$.

Further Reading

If you want to learn more about simplifying expressions and exponentiation, here are some additional resources:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Simplifying Expressions with Exponents
• Wolfram Alpha: Exponentiation and Simplification

Final Answer

The final answer is: $\boxed{1679616}$

Introduction

In our previous article, we simplified the expression $\left(6^2\right)4$ using the power of a power rule. We reviewed the basics of exponents and applied the rule to simplify the expression. In this article, we will answer some frequently asked questions related to simplifying expressions and exponentiation.

Q&A

Q: What is the power of a power rule?

A: The power of a power rule states that when we raise a power to another power, we multiply the exponents. This rule can be expressed as:

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

where $a$ is the base, $m$ and $n$ are the exponents, and the result is a new exponent.

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

A: To apply the power of a power rule, we multiply the exponents of the two powers. For example, in the expression $\left(6^2\right)4$, we multiply the exponent $2$ by the exponent $4$ to get $8$.

Q: What is the final result of the expression $\left(6^2\right)4$?

A: The final result of the expression $\left(6^2\right)4$ is $1679616$.

Q: Can we simplify expressions with negative exponents?

A: Yes, we can simplify expressions with negative exponents. When we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. For example, in the expression $\frac{1}{a^m}$, we can rewrite it as $a^{-m}$.

Q: How do we simplify expressions with fractional exponents?

A: To simplify expressions with fractional exponents, we can use the rule that $a^{m/n} = \sqrt[n]{a^m}$. For example, in the expression $\sqrt[3]{a^2}$, we can rewrite it as $a^{2/3}$.

Q: Can we simplify expressions with multiple bases?

A: Yes, we can simplify expressions with multiple bases. When we have multiple bases, we can use the rule that $a^m \cdot b^n = (ab)^{m+n}$. For example, in the expression $a^2 \cdot b^3$, we can rewrite it as $(ab)^5$.

Examples

Example 1: Simplifying an Expression with a Power of a Power

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

Solution:

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

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

Example 2: Simplifying an Expression with a Negative Exponent

Simplify the expression $\frac{1}{a^3}$.

Solution:

Using the rule that $\frac{1}{a^m} = a^{-m}$, we can rewrite the expression as follows:

$$\frac{1}{a^3} = a^{-3}$$

Example 3: Simplifying an Expression with a Fractional Exponent

Simplify the expression $\sqrt[3]{a^2}$.

Solution:

Using the rule that $a^{m/n} = \sqrt[n]{a^m}$, we can rewrite the expression as follows:

$$\sqrt[3]{a^2} = a^{2/3}$$

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions and exponentiation. We reviewed the power of a power rule and applied it to simplify expressions with multiple bases. We also discussed how to simplify expressions with negative exponents and fractional exponents. By following these steps, we can simplify complex expressions and solve problems efficiently.

Further Reading

If you want to learn more about simplifying expressions and exponentiation, here are some additional resources:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Simplifying Expressions with Exponents
• Wolfram Alpha: Exponentiation and Simplification

Final Answer

The final answer is: $\boxed{1679616}$