Write The Following Expression Without Negative Exponents And Without Parentheses. $(3x)^{-2}$

Introduction

Exponential expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will focus on simplifying the expression $(3x)^{-2}$ without negative exponents and without parentheses.

Understanding Exponents

Before we dive into simplifying the expression, let's briefly review what exponents are. An exponent is a small number that is raised to the power of a variable or a constant. In the expression $(3x)^{-2}$, the exponent $-2$ indicates that the base $(3x)$ is being raised to the power of $-2$.

Simplifying Negative Exponents

To simplify the expression $(3x)^{-2}$, we need to get rid of the negative exponent. We can do this by using the rule that states $a^{-n} = \frac{1}{a^n}$. Applying this rule to our expression, we get:

$$(3x)^{-2} = \frac{1}{(3x)^2}$$

Simplifying the Expression

Now that we have eliminated the negative exponent, we can simplify the expression further by expanding the squared term:

$$(3x)^2 = (3x)(3x) = 9x^2$$

Substituting this back into our expression, we get:

$$\frac{1}{(3x)^2} = \frac{1}{9x^2}$$

Final Answer

Therefore, the simplified expression without negative exponents and without parentheses is:

$$\frac{1}{9x^2}$$

Conclusion

In this article, we have demonstrated how to simplify the expression $(3x)^{-2}$ without negative exponents and without parentheses. By applying the rules of exponents and simplifying the expression step-by-step, we arrived at the final answer of $\frac{1}{9x^2}$. This result highlights the importance of understanding and applying the rules of exponents in mathematical problem-solving.

Common Mistakes to Avoid

When simplifying exponential expressions, it's essential to avoid common mistakes such as:

• Forgetting to eliminate negative exponents
• Failing to expand squared terms
• Not simplifying the expression further

By being aware of these potential pitfalls, you can ensure that your simplifications are accurate and reliable.

Real-World Applications

Simplifying exponential expressions has numerous real-world applications in fields such as:

• Physics: Exponential expressions are used to describe the behavior of physical systems, such as the decay of radioactive materials.
• Engineering: Exponential expressions are used to model population growth, chemical reactions, and other complex systems.
• Economics: Exponential expressions are used to model economic growth, inflation, and other economic phenomena.

By mastering the art of simplifying exponential expressions, you can apply these skills to a wide range of real-world problems and make a meaningful impact in various fields.

Additional Resources

For further practice and review, we recommend the following resources:

• Khan Academy: Exponents and Exponential Functions
• MIT OpenCourseWare: Exponents and Logarithms
• Wolfram Alpha: Exponential Expressions

Q: What is the rule for simplifying negative exponents?

A: The rule for simplifying negative exponents is $a^{-n} = \frac{1}{a^n}$. This means that to simplify a negative exponent, you can rewrite it as a fraction with the base in the denominator and the exponent as a positive number.

Q: How do I simplify an expression with a negative exponent and a squared term?

A: To simplify an expression with a negative exponent and a squared term, you can follow these steps:

1. Eliminate the negative exponent by rewriting it as a fraction with the base in the denominator and the exponent as a positive number.
2. Expand the squared term by multiplying the base by itself.
3. Simplify the resulting expression by combining like terms.

Q: What is the difference between an exponential expression and a polynomial expression?

A: An exponential expression is an expression that contains a base raised to a power, such as $2^3$ or $x^2$. A polynomial expression, on the other hand, is an expression that contains a sum of terms, each of which is a constant or a variable raised to a non-negative power, such as $2x^2 + 3x - 4$.

Q: Can I simplify an expression with a negative exponent and a fraction?

A: Yes, you can simplify an expression with a negative exponent and a fraction by following these steps:

1. Eliminate the negative exponent by rewriting it as a fraction with the base in the denominator and the exponent as a positive number.
2. Simplify the resulting fraction by combining like terms.

Q: How do I simplify an expression with a negative exponent and a variable in the denominator?

A: To simplify an expression with a negative exponent and a variable in the denominator, you can follow these steps:

1. Eliminate the negative exponent by rewriting it as a fraction with the base in the denominator and the exponent as a positive number.
2. Simplify the resulting fraction by combining like terms.

Q: What is the final answer to the expression $(3x)^{-2}$?

A: The final answer to the expression $(3x)^{-2}$ is $\frac{1}{9x^2}$.

Q: Can I use a calculator to simplify an exponential expression?

A: Yes, you can use a calculator to simplify an exponential expression. However, it's always a good idea to double-check your work by simplifying the expression by hand.

Q: How do I know if an expression is an exponential expression or a polynomial expression?

A: To determine if an expression is an exponential expression or a polynomial expression, look for the presence of a base raised to a power. If the expression contains a base raised to a power, it is an exponential expression. If the expression contains a sum of terms, each of which is a constant or a variable raised to a non-negative power, it is a polynomial expression.

Q: Can I simplify an expression with a negative exponent and a radical?

A: Yes, you can simplify an expression with a negative exponent and a radical by following these steps:

1. Eliminate the negative exponent by rewriting it as a fraction with the base in the denominator and the exponent as a positive number.
2. Simplify the resulting fraction by combining like terms.

Q: How do I simplify an expression with a negative exponent and a trigonometric function?

A: To simplify an expression with a negative exponent and a trigonometric function, you can follow these steps:

1. Eliminate the negative exponent by rewriting it as a fraction with the base in the denominator and the exponent as a positive number.
2. Simplify the resulting fraction by combining like terms.

Q: Can I use a graphing calculator to simplify an exponential expression?

A: Yes, you can use a graphing calculator to simplify an exponential expression. However, it's always a good idea to double-check your work by simplifying the expression by hand.