Problem 26 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. \(\frac{m n^{2}}{m^{-2}}\)

Step-by-Step Solution

Answer

The simplified expression is \( m^{3} n^{2} \).

Step 1: Apply the Properties of Exponents

The given expression is \( \frac{m n^{2}}{m^{-2}} \). We know that when dividing like bases, we subtract the exponents. So we start by subtracting the exponent of \( m \) in the denominator from the exponent of \( m \) in the numerator: \( 1 - (-2) = 1 + 2 = 3 \). This simplifies our expression to \( m^{3} n^{2} \).

Step 2: Ensure Positive Exponents

We notice that after simplifying, \( m^{3} n^{2} \) already has all positive exponents. No further simplification is needed.

Key Concepts

Properties of Exponents

The Properties of Exponents are crucial when simplifying expressions, especially those with variables. These properties help us manage and manipulate expressions with powers efficiently. Here are some common properties you need to remember:

• **Product of Powers:** If you have the same base, you add the exponents: \( a^m \cdot a^n = a^{m+n} \).
• **Quotient of Powers:** Again, for the same base, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• **Power of a Power:** You multiply the exponents: \( (a^m)^n = a^{m \times n} \).
• **Zero Exponent:** Any base raised to the zero power equals 1: \( a^0 = 1 \), provided \( a eq 0 \).

These properties allow us to simplify expressions with ease, as seen in the example step where the expression \( \frac{m n^{2}}{m^{-2}} \) was simplified to \( m^{3} n^{2} \) by subtracting the exponents of \( m \). This helps in reducing complex expressions to a simpler form, ensuring clarity and ease of computation.

Positive Exponents

In mathematical expressions, it's often preferred to express variables with positive exponents. This keeps the expressions clean and easy to understand.
Let's break down positive exponents:

• **Positive Exponents:** These indicate how many times you multiply the base by itself. For example, \( x^3 \) means \( x \times x \times x \).
• **Converting Negative to Positive:** If you have a negative exponent, such as \( a^{-n} \), it can be rewritten as a fraction: \( \frac{1}{a^n} \). This is because a negative exponent indicates division, not multiplication.

In our example, the expression \( m^3 n^2 \) was already expressed with positive exponents, which means no further changes were necessary. Working with positive exponents ensures that the solution is straightforward and avoids unnecessary complexity.

Algebraic Fractions

Algebraic Fractions are fractions where the numerator, the denominator, or both are algebraic expressions. Simplifying these fractions often involves applying exponent rules and factoring.
Here’s how you can manage algebraic fractions:

• **Simplification:** Use properties of exponents to simplify terms within the fraction. For instance, if you have \( \frac{m n^{2}}{m^{-2}} \), you would utilize the quotient of powers rule to simplify the expression by subtracting the exponents.
• **Factorization:** Look for common factors in the numerator and the denominator. This might not always be applicable, but it's worth checking as it can immensely simplify your fractions.
• **Rewriting:** If necessary, convert any negative exponents to positive to make understanding and further calculations easier.

By approaching algebraic fractions with these strategies in mind, you simplify the process of handling and solving expressions, transforming them into a more manageable form—as observed when simplifying the example \( \frac{m n^{2}}{m^{-2}} \) to \( m^3 n^2 \). This ensures your solutions are accurate and comprehensible.