Which Expression Is Equivalent To The Given Expression? \left(3 M^{-4}\right)^3\left(3 M^5\right ]A. 81 M 2 \frac{81}{m^2} M 2 81 ​ B. 27 M 7 \frac{27}{m^7} M 7 27 ​ C. 27 M 2 \frac{27}{m^2} M 2 27 ​ D. 81 M 7 \frac{81}{m^7} M 7 81 ​

Understanding Exponential Expressions

Exponential expressions are a fundamental concept in mathematics, and they play a crucial role in various mathematical operations. In this article, we will focus on simplifying exponential expressions, specifically those involving negative exponents and powers of variables.

The Given Expression

The given expression is $\left(3 m^{-4}\right)^3\left(3 m^5\right)$. Our goal is to simplify this expression and determine which of the given options is equivalent to it.

Simplifying the Expression

To simplify the given expression, we need to apply the rules of exponents. The first step is to expand the expression using the power of a product rule, which states that $(ab)^n = a^n b^n$.

(3 m^{-4})^3 (3 m^5) = 3^3 (m^{-4})^3 3 m^5

Next, we can simplify the expression further by applying the power of a power rule, which states that $(a^m)^n = a^{mn}$.

3^3 (m^{-4})^3 3 m^5 = 3^3 3 m^{-12} m^5

Now, we can combine the like terms and simplify the expression.

3^3 3 m^{-12} m^5 = 3^4 m^{-7}

Evaluating the Expression

To evaluate the expression, we need to calculate the value of $3^4$ and $m^{-7}$.

3^4 = 81
m^{-7} = \frac{1}{m^7}

Therefore, the simplified expression is $\frac{81}{m^7}$.

Conclusion

In conclusion, the given expression $\left(3 m^{-4}\right)^3\left(3 m^5\right)$ is equivalent to $\frac{81}{m^7}$. This is the correct answer among the given options.

Answer Key

The correct answer is:

• D. $\frac{81}{m^7}$

Why Choose This Answer?

This answer is correct because it is the result of simplifying the given expression using the rules of exponents. The other options are incorrect because they do not accurately represent the simplified expression.

Tips and Tricks

When simplifying exponential expressions, it is essential to apply the rules of exponents in the correct order. This includes expanding the expression using the power of a product rule, simplifying the expression using the power of a power rule, and combining like terms.

Common Mistakes

One common mistake when simplifying exponential expressions is to forget to apply the power of a power rule. This can lead to incorrect results and make it challenging to determine the correct answer.

Real-World Applications

Exponential expressions have numerous real-world applications, including finance, science, and engineering. Understanding how to simplify these expressions is crucial for making accurate calculations and predictions.

Final Thoughts

Q: What is the rule for simplifying exponential expressions?

A: The rule for simplifying exponential expressions involves applying the power of a product rule, the power of a power rule, and combining like terms. This process helps to simplify the expression and determine the correct answer.

Q: How do I apply the power of a product rule?

A: To apply the power of a product rule, you need to expand the expression using the formula $(ab)^n = a^n b^n$. This rule helps to simplify expressions involving multiple variables and exponents.

Q: What is the power of a power rule?

A: The power of a power rule states that $(a^m)^n = a^{mn}$. This rule helps to simplify expressions involving exponents of exponents.

Q: How do I combine like terms?

A: To combine like terms, you need to identify the terms with the same variable and exponent. You can then add or subtract these terms to simplify the expression.

Q: What is the difference between a positive and negative exponent?

A: A positive exponent indicates that the variable is raised to a power, while a negative exponent indicates that the variable is raised to a reciprocal power. For example, $x^2$ is a positive exponent, while $x^{-2}$ is a negative exponent.

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

A: To simplify an expression with a negative exponent, you need to apply the rule for negative exponents, which states that $a^{-n} = \frac{1}{a^n}$. This rule helps to simplify expressions involving negative exponents.

Q: What is the rule for multiplying exponential expressions?

A: The rule for multiplying exponential expressions states that $a^m \cdot a^n = a^{m+n}$. This rule helps to simplify expressions involving multiple exponential terms.

Q: How do I simplify an expression with multiple exponential terms?

A: To simplify an expression with multiple exponential terms, you need to apply the rule for multiplying exponential expressions and combine like terms.

Q: What is the rule for dividing exponential expressions?

A: The rule for dividing exponential expressions states that $\frac{a^m}{a^n} = a^{m-n}$. This rule helps to simplify expressions involving multiple exponential terms.

Q: How do I simplify an expression with multiple exponential terms and a division?

A: To simplify an expression with multiple exponential terms and a division, you need to apply the rule for dividing exponential expressions and combine like terms.

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

A: Some common mistakes to avoid when simplifying exponential expressions include forgetting to apply the power of a power rule, not combining like terms, and not applying the rule for negative exponents.

Q: How can I practice simplifying exponential expressions?

A: You can practice simplifying exponential expressions by working through examples and exercises, using online resources and calculators, and seeking help from a teacher or tutor.

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

A: Simplifying exponential expressions has numerous real-world applications, including finance, science, and engineering. Understanding how to simplify these expressions is crucial for making accurate calculations and predictions.

Q: Why is it important to understand how to simplify exponential expressions?

A: Understanding how to simplify exponential expressions is essential for success in mathematics and other fields. It helps to build problem-solving skills, critical thinking, and analytical abilities.