Problem 32 - Algebra 2

Question

Simplify each expression. \(\frac{\frac{m+n}{5}}{\frac{m^{2}+n^{2}}{5}}\)

Step-by-Step Solution

Answer

\( \frac{m+n}{m^{2}+n^{2}} \) is the simplified expression.

Step 1: Write the Expression

We start by writing down the given expression in a readable format: \[ \frac{\frac{m+n}{5}}{\frac{m^{2}+n^{2}}{5}} \]

Step 2: Multiply by Reciprocal

When dividing by a fraction, we multiply by its reciprocal. Therefore, the expression becomes:\[ \frac{m+n}{5} \times \frac{5}{m^{2}+n^{2}} \]

Step 3: Simplify Common Terms

We notice that the "5" in the numerator and "5" in the denominator cancel each other out, simplifying the expression:\[ \frac{m+n}{m^{2}+n^{2}} \]

Step 4: Verify Simplification

Check the resulting expression to ensure all possible simplifications have been applied. In this case, no further simplification is possible as \( m+n \) and \( m^2+n^2 \) do not have common factors.

Key Concepts

Understanding Rational Expressions

Rational expressions are like fractions but with polynomials in the numerator and the denominator. Think of fractions with whole numbers, but instead of whole numbers, we have algebraic expressions that might include variables like \( m \), \( n \), or others. Simplifying these expressions means finding the simplest form that retains the same value. For example, if we have an expression like \( \frac{m+n}{5} \), both \( m+n \) and 5 are components of a rational expression.
Key points to remember about rational expressions:

• They must have a non-zero denominator because dividing by zero is undefined.
• The expression becomes undefined if any operations make the denominator zero.
• Simplifying is done similarly to simplifying numeric fractions, by canceling common factors in numerator and denominator.

Breaking Down Algebraic Fractions

Algebraic fractions are just special types of fractions where the top (numerator) and bottom (denominator) are made up of algebraic expressions, including variables and constants. This is where algebra meets fractions: everything we know about fractions applies, but we have to keep an eye on those variables! Let's take a look at our expression \( \frac{m+n}{5} \) over \( \frac{m^2+n^2}{5} \): each segment is an algebraic fraction.
Here’s what you need to know:

• Always consider factoring both the numerator and the denominator completely, as common factors between them can simplify the expression.
• Operations such as addition, subtraction, multiplication, and division work similarly but require careful handling of variables.
• After factoring, simplify by canceling out any common factors across the numerator and the denominator of the fraction.

The Essentials of Reciprocal Multiplication

When dealing with division of fractions, reciprocal multiplication is the technique we rely on. Instead of dividing, we multiply by the reciprocal of the divisor, or the fraction you are dividing by. In simple terms, a reciprocal of a number or a fraction is what you get when you "flip" it. For example, the reciprocal of \( \frac{m^2+n^2}{5} \) is \( \frac{5}{m^2+n^2} \).
Here's how to simplify using reciprocal multiplication:

• Identify the fraction you are dividing by.
• Find the reciprocal of this fraction by swapping its numerator and denominator.
• Multiply the original fraction by this reciprocal. So, \( \frac{\frac{m+n}{5}}{\frac{m^2+n^2}{5}} \) becomes \( \frac{m+n}{5} \times \frac{5}{m^2+n^2} \).
• Proceed to simplify the resulting expression by cancelling out identical terms, yielding \( \frac{m+n}{m^2+n^2} \).