Simplify The Following Expression: 3 11 5 ÷ 3 − 9 5 3^{\frac{11}{5}} \div 3^{-\frac{9}{5}} 3 5 11 ​ ÷ 3 − 5 9 ​ A. 1 81 \frac{1}{81} 81 1 ​ B. 1 12 \frac{1}{12} 12 1 ​ C. 81D. 12

Introduction

In this article, we will simplify the given expression $3^{\frac{11}{5}} \div 3^{-\frac{9}{5}}$. This involves applying the rules of exponents and understanding the concept of division with exponents. We will break down the solution into manageable steps, making it easy to follow and understand.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $3^4$ means $3 \times 3 \times 3 \times 3$. When we have a negative exponent, it means we are taking the reciprocal of the base raised to the positive exponent. For instance, $3^{-4}$ means $\frac{1}{3^4}$.

Simplifying the Expression

To simplify the given expression, we will use the rule of division with exponents, which states that when we divide two powers with the same base, we subtract the exponents. In this case, we have:

$$3^{\frac{11}{5}} \div 3^{-\frac{9}{5}}$$

Using the rule of division with exponents, we can rewrite the expression as:

$$3^{\frac{11}{5} - (-\frac{9}{5})}$$

Simplifying the Exponents

Now, let's simplify the exponents inside the parentheses. When we subtract a negative number, it is equivalent to adding the positive number. Therefore, we can rewrite the expression as:

$$3^{\frac{11}{5} + \frac{9}{5}}$$

Combining the Exponents

To add the fractions, we need to have a common denominator. In this case, the common denominator is 5. Therefore, we can rewrite the expression as:

$$3^{\frac{20}{5}}$$

Simplifying the Fraction

Now, let's simplify the fraction inside the exponent. We can divide the numerator and denominator by their greatest common divisor, which is 5. Therefore, we can rewrite the expression as:

$$3^{4}$$

Evaluating the Expression

Finally, let's evaluate the expression. Since $3^4$ means $3 \times 3 \times 3 \times 3$, we can calculate the value as:

$$3^4 = 81$$

Conclusion

In this article, we simplified the given expression $3^{\frac{11}{5}} \div 3^{-\frac{9}{5}}$ using the rules of exponents. We applied the rule of division with exponents, simplified the exponents, combined them, and finally evaluated the expression. The final answer is $\boxed{81}$.

Common Mistakes to Avoid

When simplifying expressions with exponents, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not applying the rule of division with exponents: When dividing two powers with the same base, we need to subtract the exponents. Failing to do so can lead to incorrect results.
• Not simplifying the exponents: Exponents can be simplified by combining like terms or using the rules of exponents. Failing to simplify the exponents can lead to incorrect results.
• Not evaluating the expression: Finally, we need to evaluate the expression by applying the rules of exponents and simplifying the result. Failing to do so can lead to incorrect results.

Practice Problems

To practice simplifying expressions with exponents, try the following problems:

• $2^{\frac{7}{3}} \div 2^{-\frac{4}{3}}$
• $5^{\frac{2}{5}} \div 5^{-\frac{3}{5}}$
• $4^{\frac{3}{2}} \div 4^{-\frac{1}{2}}$

Conclusion

$$$$

Introduction

In our previous article, we simplified the given expression $3^{\frac{11}{5}} \div 3^{-\frac{9}{5}}$ using the rules of exponents. In this article, we will provide a Q&A guide to help you understand the concept better.

Q: What is the rule of division with exponents?

A: The rule of division with exponents states that when we divide two powers with the same base, we subtract the exponents. For example, $3^{\frac{11}{5}} \div 3^{-\frac{9}{5}}$ can be rewritten as $3^{\frac{11}{5} - (-\frac{9}{5})}$.

Q: How do I simplify the exponents inside the parentheses?

A: When we subtract a negative number, it is equivalent to adding the positive number. Therefore, we can rewrite the expression as $3^{\frac{11}{5} + \frac{9}{5}}$.

Q: How do I combine the exponents?

A: To add the fractions, we need to have a common denominator. In this case, the common denominator is 5. Therefore, we can rewrite the expression as $3^{\frac{20}{5}}$.

Q: How do I simplify the fraction inside the exponent?

A: We can divide the numerator and denominator by their greatest common divisor, which is 5. Therefore, we can rewrite the expression as $3^{4}$.

Q: What is the final answer to the expression $3^{\frac{11}{5}} \div 3^{-\frac{9}{5}}$?

A: The final answer is $\boxed{81}$.

Q: What are some common mistakes to avoid when simplifying expressions with exponents?

A: Some common mistakes to avoid include:

• Not applying the rule of division with exponents
• Not simplifying the exponents
• Not evaluating the expression

Q: How can I practice simplifying expressions with exponents?

A: You can practice simplifying expressions with exponents by trying the following problems:

• $2^{\frac{7}{3}} \div 2^{-\frac{4}{3}}$
• $5^{\frac{2}{5}} \div 5^{-\frac{3}{5}}$
• $4^{\frac{3}{2}} \div 4^{-\frac{1}{2}}$

Q: What are some real-world applications of simplifying expressions with exponents?

A: Simplifying expressions with exponents has many real-world applications, including:

• Calculating interest rates and investments
• Determining the area and volume of shapes
• Modeling population growth and decay

Conclusion

In this article, we provided a Q&A guide to help you understand the concept of simplifying expressions with exponents. We covered common mistakes to avoid, practice problems, and real-world applications. By following these guidelines, you can improve your skills and become more confident in simplifying expressions with exponents.

Additional Resources

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

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

Conclusion

Simplifying expressions with exponents is a fundamental concept in mathematics that has many real-world applications. By understanding the rules of exponents and practicing simplifying expressions, you can become more confident and proficient in solving problems. Remember to avoid common mistakes, practice regularly, and explore real-world applications to deepen your understanding.