Problem 79 - College Algebra with Modeling and Visualization

Question

Exercises \(65-90:\) Use rules of exponents to simplify the expression. Use positive exponents to write your answer. $$ \frac{\left(-m^{2} n^{-1}\right)^{-2}}{(m n)^{-1}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{n^3}{m^3} \).

Step 1 - Simplify the Power of a Power

Start by simplifying the numerator \( (-m^2 n^{-1})^{-2} \). According to the power of a power rule, \((a^m)^n = a^{m \cdot n}\). Therefore, \((-m^2 n^{-1})^{-2} = (-1)^{-2} \cdot m^{-4} \cdot n^2 = 1 \cdot m^{-4} \cdot n^2 = m^{-4}n^{2}\), since \((-1)^{-2} = 1\).

Step 2 - Simplify the Denominator

Simplify the denominator \( (m n)^{-1} \). Using the power of a product rule, \((ab)^n = a^n \cdot b^n\), this becomes \(m^{-1} n^{-1}\).

Step 3 - Create a Single Fraction and Apply the Division Rule

Now, simplify the expression \( \frac{m^{-4} n^{2}}{m^{-1} n^{-1}} \). Use the division rule for exponents, \( \frac{a^m}{a^n} = a^{m-n} \), for each variable: \( \frac{m^{-4}}{m^{-1}} = m^{-4 - (-1)} = m^{-3} \) and \( \frac{n^{2}}{n^{-1}} = n^{2 - (-1)} = n^{3} \).

Step 4 - Combine the Results

Combine the results from Step 3 to write the expression using positive exponents: \( m^{-3} n^{3} = \frac{n^3}{m^3} \).

Step 5 - Write the Final Expression with Positive Exponents

The simplified expression with positive exponents is \( \frac{n^3}{m^3} \).

Key Concepts

Exponent Rules

Understanding exponent rules is crucial when working with powers in algebraic expressions. These rules help simplify expressions by handling powers efficiently and accurately. Here are a few important exponent rules to keep in mind:

• Power of a Power Rule: When you raise an exponent to another power, you multiply the exponents: \((a^m)^n = a^{m imes n}\).
• Power of a Product Rule: Distribute the exponent to all parts of the product: \((ab)^n = a^n \cdot b^n\).
• Negative Exponent Rule: A negative exponent means you take the reciprocal: \(a^{-n} = \frac{1}{a^n}\).
• Division Rule: When dividing like bases, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).

By mastering these rules, you can tackle and simplify complex expressions involving exponents smoothly. Practice applying these rules, as it will significantly enhance your algebraic manipulation skills.

Simplifying Expressions

Simplifying expressions is about reducing complexity and making them easier to work with. The step-by-step approach in the original exercise demonstrates how to use exponent rules to achieve this.

Start with isolating parts of the expression you can simplify. Identify components such as products and quotients, and apply relevant exponent rules to each part.

• Identify Each Component: Break down the expression by parts, such as numerators and denominators.
• Apply Exponent Rules: Use known rules strategically. For instance, take \((-m^2 n^{-1})^{-2}\) and recognize the power and negative exponent it contains, which can be simplified using the power of a power and negative exponent rules.
• Combine Simplified Parts: After each part is simplified, combine them to form a more streamlined expression.

This systematic approach helps transform complex algebraic expressions into simpler forms that are much easier to handle and solve.

Algebraic Fractions

Algebraic fractions involve expressions that contain numerators and denominators made of algebraic expressions. Simplifying such fractions requires a strong grasp of both algebraic operations and exponent rules.

In the given exercise, the expression \(\frac{m^{-4} n^{2}}{m^{-1} n^{-1}}\) is an algebraic fraction. Here's how you can simplify it efficiently:

• Match Like Bases: Look for terms where the variable base is the same, such as the \(m\) terms or the \(n\) terms.
• Apply the Division Rule: Use the division rule for exponents where you subtract the exponents of like bases. For example, \(\frac{m^{-4}}{m^{-1}}\) involves subtracting -1 from -4, which results in \(m^{-3}\).
• Convert Negative Exponents: Change negative exponents to positive by taking the reciprocal. So, \(m^{-3}\) becomes \(\frac{1}{m^3}\).

Through these steps, you convert complex algebraic fractions into simplified forms with positive exponents, making them easier to interpret and work with. This reduces errors and lays the groundwork for further mathematical operations.