Simplify $\left(3 A^2\right)^3$.

$$

Understanding the Problem

Exponentiation is a mathematical operation that involves raising a number to a power. In this case, we are given the expression $\left(3 a^2\right)^3$, where we need to simplify the expression by applying the rules of exponentiation.

Applying the Power Rule

The power rule states that for any numbers $a$ and $b$ and any integer $n$, we have $(ab)^n = a^nb^n$. We can use this rule to simplify the expression $\left(3 a^2\right)^3$.

Step 1: Apply the Power Rule

Using the power rule, we can rewrite the expression $\left(3 a^2\right)^3$ as $3^3 (a^2)^3$. This is because the power rule states that we can separate the exponentiation of the two factors.

Step 2: Simplify the Expression

Now, we can simplify the expression $3^3 (a^2)^3$ by evaluating the exponents. We have $3^3 = 27$ and $(a^2)^3 = a^6$.

Step 3: Combine the Factors

Finally, we can combine the factors to get the simplified expression $27a^6$.

Conclusion

In this article, we have simplified the expression $\left(3 a^2\right)^3$ using the power rule of exponentiation. We have shown that the expression can be rewritten as $3^3 (a^2)^3$, and then simplified further to get the final answer $27a^6$.

Real-World Applications

The concept of exponentiation is used extensively in various fields, including science, engineering, and finance. For example, in physics, the concept of exponentiation is used to describe the behavior of physical systems, such as the growth of populations or the decay of radioactive materials. In finance, the concept of exponentiation is used to calculate compound interest and investment returns.

Tips and Tricks

• When simplifying expressions involving exponentiation, it is essential to apply the power rule correctly.
• Make sure to separate the exponentiation of the two factors using the power rule.
• Evaluate the exponents carefully to avoid errors.
• Combine the factors to get the final simplified expression.

Common Mistakes

• Failing to apply the power rule correctly.
• Not separating the exponentiation of the two factors.
• Evaluating the exponents incorrectly.
• Not combining the factors to get the final simplified expression.

Final Answer

The final answer is $\boxed{27a^6}$.

Additional Resources

For more information on exponentiation and simplifying expressions, please refer to the following resources:

• Khan Academy: Exponentiation
• Mathway: Simplifying Expressions
• Wolfram Alpha: Exponentiation

Conclusion

In conclusion, simplifying expressions involving exponentiation requires careful application of the power rule and evaluation of the exponents. By following the steps outlined in this article, you can simplify expressions like $\left(3 a^2\right)^3$ and get the final answer $27a^6$.

$$

Understanding the Problem

Exponentiation is a mathematical operation that involves raising a number to a power. In this case, we are given the expression $\left(3 a^2\right)^3$, where we need to simplify the expression by applying the rules of exponentiation.

Q&A

Q: What is the power rule of exponentiation?

A: The power rule states that for any numbers $a$ and $b$ and any integer $n$, we have $(ab)^n = a^nb^n$. This means that we can separate the exponentiation of the two factors.

Q: How do I apply the power rule to simplify the expression $\left(3 a^2\right)^3$?

A: To apply the power rule, we can rewrite the expression $\left(3 a^2\right)^3$ as $3^3 (a^2)^3$. This is because the power rule states that we can separate the exponentiation of the two factors.

Q: What is the value of $3^3$?

A: The value of $3^3$ is 27.

Q: What is the value of $(a^2)^3$?

A: The value of $(a^2)^3$ is $a^6$.

Q: How do I combine the factors to get the final simplified expression?

A: To combine the factors, we can multiply the values of $3^3$ and $(a^2)^3$. This gives us the final simplified expression $27a^6$.

Q: What are some common mistakes to avoid when simplifying expressions involving exponentiation?

A: Some common mistakes to avoid include:

• Failing to apply the power rule correctly.
• Not separating the exponentiation of the two factors.
• Evaluating the exponents incorrectly.
• Not combining the factors to get the final simplified expression.

Q: What are some real-world applications of exponentiation?

A: Exponentiation is used extensively in various fields, including science, engineering, and finance. For example, in physics, the concept of exponentiation is used to describe the behavior of physical systems, such as the growth of populations or the decay of radioactive materials. In finance, the concept of exponentiation is used to calculate compound interest and investment returns.

Q: Where can I find more information on exponentiation and simplifying expressions?

A: For more information on exponentiation and simplifying expressions, please refer to the following resources:

• Khan Academy: Exponentiation
• Mathway: Simplifying Expressions
• Wolfram Alpha: Exponentiation

Conclusion

In conclusion, simplifying expressions involving exponentiation requires careful application of the power rule and evaluation of the exponents. By following the steps outlined in this article and avoiding common mistakes, you can simplify expressions like $\left(3 a^2\right)^3$ and get the final answer $27a^6$.

Additional Resources

For more information on exponentiation and simplifying expressions, please refer to the following resources:

• Khan Academy: Exponentiation
• Mathway: Simplifying Expressions
• Wolfram Alpha: Exponentiation

Final Answer

The final answer is $\boxed{27a^6}$.

Frequently Asked Questions

Q: What is the power rule of exponentiation?

A: The power rule states that for any numbers $a$ and $b$ and any integer $n$, we have $(ab)^n = a^nb^n$.

Q: How do I apply the power rule to simplify the expression $\left(3 a^2\right)^3$?

A: To apply the power rule, we can rewrite the expression $\left(3 a^2\right)^3$ as $3^3 (a^2)^3$.

Q: What is the value of $3^3$?

A: The value of $3^3$ is 27.

Q: What is the value of $(a^2)^3$?

A: The value of $(a^2)^3$ is $a^6$.

Q: How do I combine the factors to get the final simplified expression?

A: To combine the factors, we can multiply the values of $3^3$ and $(a^2)^3$. This gives us the final simplified expression $27a^6$.

Common Mistakes

• Failing to apply the power rule correctly.
• Not separating the exponentiation of the two factors.
• Evaluating the exponents incorrectly.
• Not combining the factors to get the final simplified expression.

Final Answer

The final answer is $\boxed{27a^6}$.