Which Expression Is Equivalent To ${ \frac{\left(3 M^{-1} N 2\right) 4}{\left(2 M^{-2} N\right)^3} }$? Assume M ≠ 0 , N ≠ 0 M \neq 0, N \neq 0 M  = 0 , N  = 0 .A. 2 M 2 N 5 2 M^2 N^5 2 M 2 N 5 B. 81 M 2 N 5 8 \frac{81 M^2 N^5}{8} 8 81 M 2 N 5 ​ C. 2 M 2 N 2 2 M^2 N^2 2 M 2 N 2 D.

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Understanding Exponents and Their Rules

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Exponents are a fundamental concept in algebra, representing the repeated multiplication of a number. When dealing with exponents, it's essential to understand the rules governing their behavior, particularly when it comes to simplifying complex expressions. In this article, we will explore the process of simplifying exponents and apply this knowledge to a specific problem.

The Rules of Exponents

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Before diving into the problem, let's review the basic rules of exponents:

• Product of Powers Rule: When multiplying two powers with the same base, add their exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power Rule: When raising a power to another power, multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.
• Quotient of Powers Rule: When dividing two powers with the same base, subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.
• Zero Exponent Rule: Any non-zero number raised to the power of zero is equal to 1. For example, $a^0 = 1$.

Simplifying the Given Expression

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

$$\frac{\left(3 m^{-1} n^2\right)^4}{\left(2 m^{-2} n\right)^3}$$

To simplify this expression, we will use the quotient of powers rule, which states that when dividing two powers with the same base, subtract their exponents.

Step 1: Apply the Power of a Power Rule

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First, let's apply the power of a power rule to the numerator and denominator separately:

$$\left(3 m^{-1} n^2\right)^4 = 3^4 \cdot (m^{-1})^4 \cdot (n^2)^4$$

$$\left(2 m^{-2} n\right)^3 = 2^3 \cdot (m^{-2})^3 \cdot (n)^3$$

Step 2: Simplify the Numerator and Denominator

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Now, let's simplify the numerator and denominator separately:

$$3^4 \cdot (m^{-1})^4 \cdot (n^2)^4 = 81 \cdot m^{-4} \cdot n^8$$

$$2^3 \cdot (m^{-2})^3 \cdot (n)^3 = 8 \cdot m^{-6} \cdot n^3$$

Step 3: Apply the Quotient of Powers Rule

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Now that we have simplified the numerator and denominator, let's apply the quotient of powers rule:

$$\frac{81 \cdot m^{-4} \cdot n^8}{8 \cdot m^{-6} \cdot n^3} = \frac{81}{8} \cdot m^{-4+6} \cdot n^{8-3}$$

Step 4: Simplify the Expression

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Finally, let's simplify the expression:

$$\frac{81}{8} \cdot m^{-4+6} \cdot n^{8-3} = \frac{81}{8} \cdot m^2 \cdot n^5$$

Conclusion

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In conclusion, the expression $\frac{\left(3 m^{-1} n^2\right)^4}{\left(2 m^{-2} n\right)^3}$ is equivalent to $\frac{81 m^2 n^5}{8}$.

Discussion

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The given expression involves simplifying exponents using the quotient of powers rule. This rule states that when dividing two powers with the same base, subtract their exponents. By applying this rule, we can simplify complex expressions and arrive at the final answer.

Final Answer

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The final answer is $\boxed{\frac{81 m^2 n^5}{8}}$.

=====================================================

Understanding Exponents and Their Rules

---

Exponents are a fundamental concept in algebra, representing the repeated multiplication of a number. When dealing with exponents, it's essential to understand the rules governing their behavior, particularly when it comes to simplifying complex expressions. In this article, we will explore the process of simplifying exponents and apply this knowledge to a specific problem.

The Rules of Exponents

---

Before diving into the problem, let's review the basic rules of exponents:

• Product of Powers Rule: When multiplying two powers with the same base, add their exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power Rule: When raising a power to another power, multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.
• Quotient of Powers Rule: When dividing two powers with the same base, subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.
• Zero Exponent Rule: Any non-zero number raised to the power of zero is equal to 1. For example, $a^0 = 1$.

Simplifying the Given Expression

---

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

$$\frac{\left(3 m^{-1} n^2\right)^4}{\left(2 m^{-2} n\right)^3}$$

To simplify this expression, we will use the quotient of powers rule, which states that when dividing two powers with the same base, subtract their exponents.

Q&A: Simplifying Exponents

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Q: What is the product of powers rule?

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

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you can use the quotient of powers rule, which states that when dividing two powers with the same base, subtract their exponents.

Q: What is the power of a power rule?

A: The power of a power rule states that when raising a power to another power, multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.

Q: What is the zero exponent rule?

A: The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. For example, $a^0 = 1$.

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

A: To apply the quotient of powers rule, you need to subtract the exponents of the two powers with the same base. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Example Problems

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Problem 1: Simplify the expression $\frac{\left(2 x^3 y^2\right)^4}{\left(3 x^2 y\right)^3}$

To simplify this expression, we will use the quotient of powers rule:

$$\frac{\left(2 x^3 y^2\right)^4}{\left(3 x^2 y\right)^3} = \frac{2^4 \cdot (x^3)^4 \cdot (y^2)^4}{3^3 \cdot (x^2)^3 \cdot (y)^3}$$

$$= \frac{16 \cdot x^{12} \cdot y^8}{27 \cdot x^6 \cdot y^3}$$

$$= \frac{16}{27} \cdot x^{12-6} \cdot y^{8-3}$$

$$= \frac{16}{27} \cdot x^6 \cdot y^5$$

Problem 2: Simplify the expression $\frac{\left(3 x^{-2} y^3\right)^4}{\left(2 x^{-1} y^2\right)^3}$

To simplify this expression, we will use the quotient of powers rule:

$$\frac{\left(3 x^{-2} y^3\right)^4}{\left(2 x^{-1} y^2\right)^3} = \frac{3^4 \cdot (x^{-2})^4 \cdot (y^3)^4}{2^3 \cdot (x^{-1})^3 \cdot (y^2)^3}$$

$$= \frac{81 \cdot x^{-8} \cdot y^{12}}{8 \cdot x^{-3} \cdot y^6}$$

$$= \frac{81}{8} \cdot x^{-8+3} \cdot y^{12-6}$$

$$= \frac{81}{8} \cdot x^{-5} \cdot y^6$$

Conclusion

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In conclusion, simplifying exponents using the quotient of powers rule is a crucial concept in algebra. By understanding the rules of exponents and applying them to complex expressions, we can arrive at the final answer.

Final Answer

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The final answer is $\boxed{\frac{81 m^2 n^5}{8}}$.