Select The Correct Answer.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 ​

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Understanding Exponents and Their Rules

Exponents are a fundamental concept in mathematics, and understanding their rules is crucial for simplifying complex expressions. In this article, we will explore the rules of exponents and apply them to simplify a given expression.

The Rules of Exponents

Before we dive into the solution, let's review the rules of exponents:

  • Product of Powers Rule: When multiplying two powers with the same base, add their exponents. For example, am⋅an=am+na^m \cdot a^n = a^{m+n}.
  • Power of a Power Rule: When raising a power to another power, multiply their exponents. For example, (am)n=am⋅n(a^m)^n = a^{m \cdot n}.
  • Quotient of Powers Rule: When dividing two powers with the same base, subtract their exponents. For example, aman=am−n\frac{a^m}{a^n} = a^{m-n}.

Simplifying the Given Expression

Now that we have reviewed the rules of exponents, let's apply them to simplify the given expression:

(3m−4)3(3m5)\left(3 m^{-4}\right)^3\left(3 m^5\right)

Using the Power of a Power Rule, we can simplify the first part of the expression:

(3m−4)3=33⋅m−4⋅3=27⋅m−12\left(3 m^{-4}\right)^3 = 3^3 \cdot m^{-4 \cdot 3} = 27 \cdot m^{-12}

Now, let's simplify the second part of the expression:

3m53 m^5

We can leave this part as is for now.

Combining the Simplified Expressions

Now that we have simplified both parts of the expression, let's combine them:

27⋅m−12⋅3m527 \cdot m^{-12} \cdot 3 m^5

Using the Product of Powers Rule, we can combine the two powers with the same base:

27⋅3⋅m−12⋅m5=81⋅m−727 \cdot 3 \cdot m^{-12} \cdot m^5 = 81 \cdot m^{-7}

Simplifying the Final Expression

Finally, let's simplify the final expression:

81⋅m−781 \cdot m^{-7}

Using the Quotient of Powers Rule, we can rewrite this expression as:

81m7\frac{81}{m^7}

Conclusion

In this article, we have applied the rules of exponents to simplify a given expression. We have used the Power of a Power Rule, Product of Powers Rule, and Quotient of Powers Rule to simplify the expression step by step. The final simplified expression is:

81m7\frac{81}{m^7}

This expression is equivalent to the given expression, and it is the correct answer.

Answer Key

The correct answer is:

D. 81m7\frac{81}{m^7}

Discussion

Q: What is the rule for simplifying exponential expressions?

A: The rule for simplifying exponential expressions is based on the following rules:

  • Product of Powers Rule: When multiplying two powers with the same base, add their exponents. For example, am⋅an=am+na^m \cdot a^n = a^{m+n}.
  • Power of a Power Rule: When raising a power to another power, multiply their exponents. For example, (am)n=am⋅n(a^m)^n = a^{m \cdot n}.
  • Quotient of Powers Rule: When dividing two powers with the same base, subtract their exponents. For example, aman=am−n\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, you can rewrite it as a fraction with a positive exponent. For example:

a−m=1ama^{-m} = \frac{1}{a^m}

Q: What is the difference between a power and an exponent?

A: A power is the result of raising a number to a certain exponent. For example, ama^m is a power, and mm is the exponent. An exponent is a number that is raised to a certain power.

Q: Can I simplify an expression with a variable in the exponent?

A: Yes, you can simplify an expression with a variable in the exponent. For example:

am+n=am⋅ana^{m+n} = a^m \cdot a^n

Q: How do I simplify an expression with a fraction in the exponent?

A: To simplify an expression with a fraction in the exponent, you can rewrite it as a product of powers. For example:

amn=(am)1na^{\frac{m}{n}} = (a^m)^{\frac{1}{n}}

Q: Can I simplify an expression with a negative base?

A: Yes, you can simplify an expression with a negative base. For example:

(−a)m=am(-a)^m = a^m if mm is even (−a)m=−am(-a)^m = -a^m if mm is odd

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

A: To simplify an expression with a zero exponent, you can rewrite it as 1. For example:

a0=1a^0 = 1

Q: Can I simplify an expression with a variable base and exponent?

A: Yes, you can simplify an expression with a variable base and exponent. For example:

am+n=am⋅ana^{m+n} = a^m \cdot a^n

Q: How do I simplify an expression with a coefficient in the exponent?

A: To simplify an expression with a coefficient in the exponent, you can rewrite it as a product of powers. For example:

akm=(am)ka^{km} = (a^m)^k

Conclusion

In this article, we have answered some frequently asked questions about simplifying exponential expressions. We have covered topics such as the rules for simplifying exponential expressions, simplifying expressions with negative exponents, and simplifying expressions with variables in the exponent. We hope that this article has been helpful in clarifying any questions you may have had about simplifying exponential expressions.

Additional Resources

If you are looking for additional resources on simplifying exponential expressions, we recommend checking out the following websites:

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

Discussion

Do you have any questions about simplifying exponential expressions? Do you want to practice simplifying more expressions? Let's discuss in the comments below!