Simplify The Expression: $\frac{\left(3 M^{-2} N\right)^{-3}}{6 M N^{-2}}$

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Let's break down how to simplify the algebraic expression $\frac{\left(3 m^{-2} n\right)^{-3}}{6 m n^{-2}}$. This involves understanding exponent rules and applying them step-by-step. We'll make sure each step is clear so you can follow along easily!

Understanding the Expression

Before diving into the simplification, let's identify what we're working with. We have a fraction where the numerator contains a term raised to a negative exponent, and the denominator is a simple algebraic expression. Our goal is to simplify this into one of the provided options.

Step-by-Step Simplification

Step 1: Simplify the Numerator

We start by simplifying the numerator, which is $\left(3 m^{-2} n\right)^{-3}$. Remember, when you raise a product to a power, you apply the power to each factor. Also, a negative exponent means you take the reciprocal of the base raised to the positive exponent.

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

Now, let's simplify each term:

• $3^{-3} = \frac{1}{3^3} = \frac{1}{27}$
• $(m^{-2})^{-3} = m^{(-2 \cdot -3)} = m^6$
• $n^{-3} = \frac{1}{n^3}$

So, the numerator becomes:

$\frac{1}{27} \cdot m^6 \cdot \frac{1}{n^3} = \frac{m^6}{27n^3}$

Step 2: Rewrite the Original Expression

Now that we've simplified the numerator, let's rewrite the original expression:

$\frac{\left(3 m^{-2} n\right)^{-3}}{6 m n^{-2}} = \frac{\frac{m^6}{27n^3}}{6 m n^{-2}}$

This looks a bit complex, but we can simplify it further by understanding that dividing by a fraction is the same as multiplying by its reciprocal.

Step 3: Simplify the Entire Fraction

We can rewrite the expression as:

$\frac{m^6}{27n^3} \div (6 m n^{-2}) = \frac{m^6}{27n^3} \cdot \frac{1}{6 m n^{-2}}$

Now, multiply the fractions:

$\frac{m^6}{27n^3} \cdot \frac{1}{6 m n^{-2}} = \frac{m^6}{27 \cdot 6 \cdot m \cdot n^3 \cdot n^{-2}} = \frac{m^6}{162 m n^{3-2}} = \frac{m^6}{162 m n}$

Step 4: Reduce the Expression

Now, we reduce the expression by canceling out common terms. We have $m^6$ in the numerator and $m$ in the denominator, so we can cancel out one $m$ from both:

$\frac{m^6}{162 m n} = \frac{m^{6-1}}{162 n} = \frac{m^5}{162 n}$

Conclusion

After simplifying the given expression $\frac{\left(3 m^{-2} n\right)^{-3}}{6 m n^{-2}}$, we arrive at $\frac{m^5}{162 n}$. This matches option A.

Therefore, the equivalent expression is:

$\frac{m^5}{162 n}$

Detailed Explanation of Exponent Rules

To truly master simplifying expressions like these, it's important to have a solid grasp of exponent rules. Let’s delve deeper into each rule applied in our simplification process.

Rule 1: Negative Exponents

A negative exponent indicates that you should take the reciprocal of the base. Mathematically, this is represented as:

$a^{-n} = \frac{1}{a^n}$

In our problem, we encountered terms like $3^{-3}$ and $n^{-3}$. Applying the rule:

• $3^{-3} = \frac{1}{3^3} = \frac{1}{27}$
• $n^{-3} = \frac{1}{n^3}$

This rule is crucial because it helps transform expressions with negative exponents into more manageable forms, allowing for easier simplification.

Rule 2: Power of a Power

When you raise a power to another power, you multiply the exponents. This is represented as:

$(a^m)^n = a^{m \cdot n}$

In our expression, we had $(m^{-2})^{-3}$. Applying this rule:

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

Understanding this rule is essential for simplifying terms where exponents are nested, making the expression easier to work with.

Rule 3: Power of a Product

When a product is raised to a power, each factor in the product is raised to that power. This is represented as:

$(ab)^n = a^n b^n$

In our problem, we applied this rule to $\left(3 m^{-2} n\right)^{-3}$:

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

This rule allows us to distribute the exponent across all terms within the parentheses, breaking down the complex expression into simpler components.

Rule 4: Dividing Powers with the Same Base

When dividing powers with the same base, you subtract the exponents. This is represented as:

$\frac{a^m}{a^n} = a^{m-n}$

In our final simplification, we had $\frac{m^6}{m}$. Applying this rule:

$\frac{m^6}{m} = m^{6-1} = m^5$

This rule helps in reducing expressions by canceling out common factors, leading to a simplified result.

Common Mistakes to Avoid

When simplifying algebraic expressions with exponents, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid errors and improve your accuracy.

Mistake 1: Incorrectly Applying Negative Exponents

A frequent mistake is misunderstanding how negative exponents work. Remember, a negative exponent means taking the reciprocal of the base, not making the base negative.

• Correct: $a^{-n} = \frac{1}{a^n}$
• Incorrect: $a^{-n} \neq -a^n$

For example, $3^{-2}$ is $\frac{1}{3^2} = \frac{1}{9}$, not $-9$.

Mistake 2: Forgetting to Distribute Exponents

When raising a product to a power, it’s essential to apply the power to each factor. Forgetting to do so can lead to incorrect simplifications.

• Correct: $(ab)^n = a^n b^n$
• Incorrect: $(ab)^n \neq a^n b$

For example, $(2x)^3 = 2^3 x^3 = 8x^3$, not $2x^3$.

Mistake 3: Adding Exponents When Multiplying Different Bases

Exponents can only be added when multiplying powers with the same base. Multiplying powers with different bases requires keeping the bases separate.

• Correct: $a^m \cdot a^n = a^{m+n}$
• Incorrect: $a^m \cdot b^n \neq (ab)^{m+n}$

For example, $2^3 \cdot 2^2 = 2^{3+2} = 2^5 = 32$, but $2^3 \cdot 3^2$ cannot be simplified by adding exponents.

Mistake 4: Not Simplifying Completely

Always ensure that your final answer is in the simplest form. This means reducing fractions, canceling out common factors, and combining like terms.

For example, if you end up with $\frac{4x^2}{2x}$, simplify it to $2x$.

Mistake 5: Confusing Power of a Power with Multiplying Powers

It’s important to differentiate between raising a power to a power and multiplying powers with the same base.

• Power of a Power: $(a^m)^n = a^{m \cdot n}$
• Multiplying Powers: $a^m \cdot a^n = a^{m+n}$

For example, $(x^2)^3 = x^{2 \cdot 3} = x^6$, while $x^2 \cdot x^3 = x^{2+3} = x^5$.

Mistake 6: Ignoring the Order of Operations

Always follow the order of operations (PEMDAS/BODMAS) when simplifying expressions. This ensures that you perform operations in the correct sequence.

By keeping these common mistakes in mind, you can approach simplifying algebraic expressions with greater confidence and accuracy. Practice and careful attention to detail are key to mastering these concepts.

To deepen your understanding of algebraic expressions and exponent rules, consider exploring resources like Khan Academy's Algebra Section. This can provide additional practice and insights.