Problem 16 - Elementary Algebra

Question

For exercises 1-66, simplify. $$ \frac{5 m^{2}+30 m}{75 m} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{m+6}{15} \).

Step 1: Factor the Numerator

Identify common factors in the numerator. In this case, both terms in the numerator share a common factor of 5m: \[ 5m(m + 6) \]. So, the expression becomes \[ \frac{5m(m + 6)}{75m} \].

Step 2: Simplify the Fraction

Cancel out the common factor in the numerator and the denominator. The numerator and the denominator both have a 5m term. Dividing these terms out, the fraction simplifies to \[ \frac{m + 6}{15} \].

Key Concepts

Simplifying Fractions with Algebraic Expressions

When simplifying algebraic fractions, you're essentially doing the same task as simplifying numerical fractions. Breaking down a complex expression into smaller, more manageable parts can make it easier to understand. The exercise we are looking at simplifies the fraction \[ \frac{5m^2 + 30m}{75m} \]. Simplification involves two main strategies: factoring and canceling out common factors. By first factoring the numerator and then simplifying by canceling out similar terms in the numerator and denominator, we can make the expression easier to handle!

Factoring

Factoring is a method where we break down an expression into a product of simpler expressions that can be multiplied together to get the original expression. For instance, in our example, \[ 5m^2 + 30m \] can be factored by taking out the common term. Both terms, \[ 5m^2 \] and \[ 30m \], have a common factor of \[ 5m \]. So, by factoring we get \[ 5m(m + 6) \].

Factoring is important because it allows us to simplify expressions by breaking them down. Here are some tips for effective factoring:

• Always look for the greatest common factor (GCF) first.
• Break down each term in the expression if needed to find the GCF.
• Rewrite the expression using the GCF.

Common Factors and Simplifying Fractions

A common factor is an expression that divides exactly into two or more expressions. In our case, the GCF of both the numerator (\[ 5m(m + 6) \]) and the denominator (\[ 75m \]) of the fraction is \[ 5m \].

To simplify the fraction, we can cancel out these common terms. This step essentially reduces the fraction by dividing both the numerator and the denominator by the GCF.

Our simplified expression is then \[ \frac{m+6}{15} \], since \[ 5m \] cancels out. Remember:

• Canceling out common factors helps to make complex fractions simpler.
• Always perform the division of the common factor to both parts of the fraction.
• Cross-check to ensure there are no more common factors left.