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

Mar 13, 2025 by ADMIN 234 views

**Simplifying Exponential Expressions: A Step-by-Step Guide**

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 the given expression $\left(3 m^{-4}\right)^3\left(3 m^5\right)$ . We will explore the properties of exponents and learn how to simplify complex exponential expressions.

Properties of Exponents

Before we dive into the given expression, let's review the properties of exponents. The properties of exponents are as follows:

Product of Powers Property: When multiplying two powers with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$ .
Power of a Power Property: When raising a power to another power, we multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$ .
Quotient of Powers Property: When dividing two powers with the same base, we subtract the exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$ .

Simplifying the Given Expression

Now that we have reviewed the properties of exponents, let's simplify the given expression $\left(3 m^{-4}\right)^3\left(3 m^5\right)$ . To simplify this expression, we will use the product of powers property and the power of a power property.

First, let's simplify the expression inside the parentheses:

\left(3 m^{-4}\right)^3 = 3^3 \cdot (m^{-4})^3 = 27 \cdot m^{-12}

Next, let's simplify the expression outside the parentheses:

3 m^5

Now, let's multiply the two simplified expressions:

27 \cdot m^{-12} \cdot 3 m^5 = 81 \cdot m^{-7}

Evaluating the Answer Choices

Now that we have simplified the given expression, let's evaluate the answer choices:

A. $\frac{81}{m^2}$ : This expression is not equivalent to the given expression.
B. $\frac{27}{m^7}$ : This expression is not equivalent to the given expression.
C. $\frac{27}{m^2}$ : This expression is not equivalent to the given expression.
D. $\frac{81}{m^7}$ : This expression is equivalent to the given expression.

Conclusion

In conclusion, the given expression $\left(3 m^{-4}\right)^3\left(3 m^5\right)$ is equivalent to the expression $\frac{81}{m^7}$ . We simplified the given expression using the product of powers property and the power of a power property. We also evaluated the answer choices and found that only one expression is equivalent to the given expression.

Tips and Tricks

Here are some tips and tricks to help you simplify exponential expressions:

Use the product of powers property to simplify expressions with the same base.
Use the power of a power property to simplify expressions with a power raised to another power.
Use the quotient of powers property to simplify expressions with a quotient of powers.
Simplify expressions inside parentheses first.
Multiply expressions with the same base by adding the exponents.
Raise a power to another power by multiplying the exponents.

By following these tips and tricks, you will be able to simplify complex exponential expressions with ease.

Common Mistakes to Avoid

Here are some common mistakes to avoid when simplifying exponential expressions:

Forgetting to use the product of powers property when multiplying expressions with the same base.
Forgetting to use the power of a power property when raising a power to another power.
Forgetting to use the quotient of powers property when dividing expressions with the same base.
Not simplifying expressions inside parentheses first.
Not multiplying expressions with the same base by adding the exponents.
Not raising a power to another power by multiplying the exponents.

By avoiding these common mistakes, you will be able to simplify exponential expressions accurately and efficiently.

Real-World Applications

Exponential expressions have many real-world applications, including:

Physics: Exponential expressions are used to describe the behavior of physical systems, such as the motion of objects and the decay of radioactive materials.
Engineering: Exponential expressions are used to describe the behavior of complex systems, such as electrical circuits and mechanical systems.
Economics: Exponential expressions are used to describe the behavior of economic systems, such as the growth of populations and the decay of resources.
Biology: Exponential expressions are used to describe the behavior of biological systems, such as the growth of populations and the decay of genetic material.

By understanding exponential expressions, you will be able to analyze and solve complex problems in various fields.

Final Thoughts

Q: What is the product of powers property?

A: The product of powers property states that when multiplying two powers with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$ .

Q: How do I simplify an expression with a power raised to another power?

A: To simplify an expression with a power raised to another power, we multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$ .

Q: What is the quotient of powers property?

A: The quotient of powers property states that when dividing two powers with the same base, we subtract the exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$ .

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

A: To simplify an expression with a negative exponent, we can rewrite it as a fraction with a positive exponent. For example, $a^{-m} = \frac{1}{a^m}$ .

Q: What is the rule for multiplying exponential expressions with the same base?

A: When multiplying exponential expressions with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$ .

Q: What is the rule for dividing exponential expressions with the same base?

A: When dividing exponential expressions with the same base, we subtract the exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$ .

Q: How do I simplify an expression with a power of a power?

A: To simplify an expression with a power of a power, we multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$ .

Q: What is the rule for raising a power to another power?

A: When raising a power to another power, we multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$ .

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

A: To simplify an expression with a negative exponent and a power of a power, we can rewrite it as a fraction with a positive exponent and multiply the exponents. For example, $(a^{-m})^n = \frac{1}{a^{m \cdot n}}$ .

Q: What is the rule for simplifying exponential expressions with the same base?

A: When simplifying exponential expressions with the same base, we can use the product of powers property, the power of a power property, and the quotient of powers property to simplify the expression.

Q: How do I apply the rules for simplifying exponential expressions?

A: To apply the rules for simplifying exponential expressions, we need to identify the properties of exponents that apply to the expression and use them to simplify the expression.

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 use the product of powers property when multiplying expressions with the same base.
Forgetting to use the power of a power property when raising a power to another power.
Forgetting to use the quotient of powers property when dividing expressions with the same base.
Not simplifying expressions inside parentheses first.
Not multiplying expressions with the same base by adding the exponents.
Not raising a power to another power by multiplying the exponents.

Q: How do I practice simplifying exponential expressions?

A: To practice simplifying exponential expressions, you can try the following:

Start with simple expressions and gradually move on to more complex expressions.
Use online resources and practice problems to help you practice simplifying exponential expressions.
Work with a partner or tutor to help you understand the concepts and practice simplifying exponential expressions.
Take your time and be patient with yourself as you practice simplifying exponential expressions.

By following these tips and practicing regularly, you will become proficient in simplifying exponential expressions and solving complex problems in various fields.