Which Value Is Equivalent To ( 3 6 ) 3 \left(3^6\right)^3 ( 3 6 ) 3 ?A. 3 2 3^2 3 2 B. 3 3 3^3 3 3 C. 3 9 3^9 3 9 D. 3 18 3^{18} 3 18

Introduction to Exponents

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. In this article, we will explore the concept of exponents and simplify the expression $\left(3^6\right)^3$ to determine its equivalent value.

Understanding the Order of Operations

When simplifying expressions with exponents, it is essential to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Simplifying $\left(3^6\right)^3$

To simplify the expression $\left(3^6\right)^3$, we need to apply the rule of exponents that states $(a^m)^n = a^{m \cdot n}$. In this case, $a = 3$, $m = 6$, and $n = 3$.

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

$\left(3^6\right)^3 = 3^{6 \cdot 3} = 3^{18}$

Evaluating the Options

Now that we have simplified the expression $\left(3^6\right)^3$ to $3^{18}$, let's evaluate the options:

A. $3^2$ B. $3^3$ C. $3^9$ D. $3^{18}$

Based on our simplification, we can see that option D, $3^{18}$, is the correct equivalent value.

Conclusion

In this article, we explored the concept of exponents and simplified the expression $\left(3^6\right)^3$ to determine its equivalent value. By applying the rule of exponents and following the order of operations, we arrived at the correct answer, $3^{18}$. This demonstrates the importance of understanding and applying exponent rules in mathematics.

Frequently Asked Questions

• Q: What is the rule of exponents for $(a^m)^n$? A: The rule of exponents states that $(a^m)^n = a^{m \cdot n}$.
• Q: How do I simplify expressions with exponents? A: To simplify expressions with exponents, follow the order of operations (PEMDAS) and apply the rule of exponents.
• Q: What is the equivalent value of $\left(3^6\right)^3$? A: The equivalent value of $\left(3^6\right)^3$ is $3^{18}$.

Additional Resources

For more information on exponents and simplifying expressions, check out the following resources:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram MathWorld: Exponents and Exponential Functions
  

Introduction to Exponents Q&A

In our previous article, we explored the concept of exponents and simplified the expression $\left(3^6\right)^3$ to determine its equivalent value. In this article, we will continue to answer frequently asked questions about exponents and provide additional resources for further learning.

Q&A: Exponents and Exponential Functions

Q: What is the rule of exponents for $(a^m)^n$?

A: The rule of exponents states that $(a^m)^n = a^{m \cdot n}$. This means that when you raise a power to a power, you multiply the exponents.

Q: How do I simplify expressions with exponents?

A: To simplify expressions with exponents, follow the order of operations (PEMDAS):

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: What is the equivalent value of $\left(3^6\right)^3$?

A: The equivalent value of $\left(3^6\right)^3$ is $3^{18}$. This can be simplified using the rule of exponents: $\left(3^6\right)^3 = 3^{6 \cdot 3} = 3^{18}$.

Q: How do I evaluate expressions with negative exponents?

A: To evaluate expressions with negative exponents, use the rule that $a^{-n} = \frac{1}{a^n}$. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Q: What is the difference between a power and an exponent?

A: A power is the result of raising a number to a power, while an exponent is the number that is being raised to a power. For example, in the expression $2^3$, 2 is the base and 3 is the exponent.

Q: How do I simplify expressions with fractional exponents?

A: To simplify expressions with fractional exponents, use the rule that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. For example, $2^{\frac{1}{2}} = \sqrt{2}$.

Additional Resources

For more information on exponents and simplifying expressions, check out the following resources:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram MathWorld: Exponents and Exponential Functions
• MIT OpenCourseWare: Exponents and Exponential Functions
• Purplemath: Exponents and Exponential Functions

Practice Problems

Try these practice problems to test your understanding of exponents and simplifying expressions:

1. Simplify the expression $\left(2^4\right)^2$.
2. Evaluate the expression $3^{-2}$.
3. Simplify the expression $\sqrt[3]{2^6}$.
4. Evaluate the expression $4^{\frac{1}{2}}$.

Conclusion

In this article, we answered frequently asked questions about exponents and provided additional resources for further learning. We also provided practice problems to test your understanding of exponents and simplifying expressions. Remember to follow the order of operations and apply the rule of exponents to simplify expressions with exponents.