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

Introduction

Exponential expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will focus on simplifying the expression $3^{\frac{11}{5}} \div 3^{-\frac{9}{5}}$. We will break down the problem step by step, using the properties of exponents to arrive at the final solution.

Understanding Exponents

Before we dive into the problem, let's review the basics of exponents. An exponent is a small number that is raised to a power, indicating how many times the base number should be multiplied by itself. For example, $3^4$ means $3 \times 3 \times 3 \times 3$. Exponents can also be negative, which means the base number is divided by itself a certain number of times.

The Quotient of Powers Property

One of the most important properties of exponents is the quotient of powers property, which states that when we divide two powers with the same base, we can subtract the exponents. Mathematically, this can be represented as:

$$\frac{a^m}{a^n} = a^{m-n}$$

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

Applying the Quotient of Powers Property

Now that we have reviewed the quotient of powers property, let's apply it to the given expression:

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

Using the quotient of powers property, we can rewrite the expression as:

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

Simplifying the Exponents

Now that we have applied the quotient of powers property, let's simplify the exponents:

$$\frac{11}{5} - (-\frac{9}{5}) = \frac{11}{5} + \frac{9}{5} = \frac{20}{5} = 4$$

So, the expression simplifies to:

$$3^4$$

Evaluating the Expression

Now that we have simplified the expression, let's evaluate it:

$$3^4 = 3 \times 3 \times 3 \times 3 = 81$$

Conclusion

In this article, we have simplified the expression $3^{\frac{11}{5}} \div 3^{-\frac{9}{5}}$ using the quotient of powers property. We have broken down the problem step by step, using the properties of exponents to arrive at the final solution. The correct answer is $\boxed{81}$.

Common Mistakes to Avoid

When simplifying exponential expressions, there are several common mistakes to avoid:

• Not applying the quotient of powers property: This is one of the most common mistakes when simplifying exponential expressions. Make sure to apply the quotient of powers property when dividing two powers with the same base.
• Not simplifying the exponents: Make sure to simplify the exponents before evaluating the expression.
• Not evaluating the expression: Make sure to evaluate the expression after simplifying the exponents.

Practice Problems

To practice simplifying exponential expressions, try the following problems:

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

Conclusion

Introduction

In our previous article, we discussed how to simplify exponential expressions using the quotient of powers property. In this article, we will provide a Q&A guide to help you understand and apply the concepts.

Q: What is the quotient of powers property?

A: The quotient of powers property states that when we divide two powers with the same base, we can subtract the exponents. Mathematically, this can be represented as:

$$\frac{a^m}{a^n} = a^{m-n}$$

Q: How do I apply the quotient of powers property?

A: To apply the quotient of powers property, follow these steps:

1. Identify the base and the exponents in the expression.
2. Check if the bases are the same.
3. If the bases are the same, subtract the exponents.
4. Simplify the resulting expression.

Q: What if the exponents are fractions?

A: If the exponents are fractions, you can still apply the quotient of powers property. To do this, follow these steps:

1. Subtract the numerators (the numbers on top of the fractions).
2. Subtract the denominators (the numbers on the bottom of the fractions).
3. Simplify the resulting expression.

Q: Can I apply the quotient of powers property to negative exponents?

A: Yes, you can apply the quotient of powers property to negative exponents. To do this, follow these steps:

1. Rewrite the negative exponent as a positive exponent with a negative base.
2. Apply the quotient of powers property.
3. Simplify the resulting expression.

Q: What if I have a fraction with a negative exponent in the denominator?

A: If you have a fraction with a negative exponent in the denominator, you can still apply the quotient of powers property. To do this, follow these steps:

1. Rewrite the fraction with a positive exponent in the denominator.
2. Apply the quotient of powers property.
3. Simplify the resulting expression.

Q: Can I apply the quotient of powers property to expressions with different bases?

A: No, you cannot apply the quotient of powers property to expressions with different bases. The quotient of powers property only applies to expressions with the same base.

Q: What if I have an expression with a variable in the exponent?

A: If you have an expression with a variable in the exponent, you can still apply the quotient of powers property. To do this, follow these steps:

1. Identify the base and the variable in the exponent.
2. Check if the bases are the same.
3. If the bases are the same, subtract the exponents.
4. Simplify the resulting expression.

Q: Can I apply the quotient of powers property to expressions with exponents with different signs?

A: Yes, you can apply the quotient of powers property to expressions with exponents with different signs. To do this, follow these steps:

1. Identify the base and the exponents with different signs.
2. Check if the bases are the same.
3. If the bases are the same, subtract the exponents.
4. Simplify the resulting expression.

Conclusion

Simplifying exponential expressions is an essential skill for students and professionals alike. By applying the quotient of powers property and understanding the concepts, you can arrive at the final solution. Remember to avoid common mistakes, such as not applying the quotient of powers property, not simplifying the exponents, and not evaluating the expression. With practice, you will become proficient in simplifying exponential expressions.

Practice Problems

To practice simplifying exponential expressions, try the following problems:

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

Common Mistakes to Avoid

When simplifying exponential expressions, there are several common mistakes to avoid:

• Not applying the quotient of powers property: This is one of the most common mistakes when simplifying exponential expressions. Make sure to apply the quotient of powers property when dividing two powers with the same base.
• Not simplifying the exponents: Make sure to simplify the exponents before evaluating the expression.
• Not evaluating the expression: Make sure to evaluate the expression after simplifying the exponents.

Conclusion

Simplifying exponential expressions is an essential skill for students and professionals alike. By applying the quotient of powers property and understanding the concepts, you can arrive at the final solution. Remember to avoid common mistakes, such as not applying the quotient of powers property, not simplifying the exponents, and not evaluating the expression. With practice, you will become proficient in simplifying exponential expressions.