Problem 57 - Algebra 1

Question

Simplify the expression. (Review \(8.3 \text { for } 11.7)\) $$\frac{4 m^{2}}{6 m}$$

Step-by-Step Solution

Answer

The expression simplifies to \(2m/3\)

Step 1: Identify the Numerator and the Denominator

Here, the numerator is \(4m^{2}\) and the denominator is \(6m\)

Step 2: Simplify the Constants

We simplify the constants \(4\) and \(6\) by dividing both by the highest common factor, which is \(2\), yielding \(2m^{2}/3m\)

Step 3: Apply Properties of Exponents

Applying the properties of exponents, we bring down the exponent of \(m\) in the numerator to \(1\), as we divide \(m^{2}\) by \(m\), leaving with \(2m/3\)

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial, especially when simplifying expressions like \( \frac{4m^2}{6m} \). Exponents are essentially a way to express repeated multiplication. When you have a base raised to a power, like \( m^2 \), it indicates that the base, \( m \), is multiplied by itself. In algebraic simplification, one of the most important rules is the division property of exponents.

Here's a quick rundown of this rule:

• When dividing similar bases by each other, you can subtract the exponent in the denominator from the exponent in the numerator.
• For example, if you have \( m^2 / m^1 \), you can subtract the exponent in the denominator from the numerator: \( 2 - 1 = 1 \).
• So \( m^2 / m \) simplifies to \( m^{2-1} \), which is \( m^1 \) or just \( m \).

Using this principle, in our expression \( 4m^2 / 6m \), the \( m \) terms simplify using exponents, and you end up with \( 2m / 3 \). Understanding and mastering these properties of exponents helps in many algebraic simplifications.

Numerator and Denominator

In the context of fractions, recognizing and understanding how to manipulate the numerator and denominator is fundamental. Consider the fraction \( \frac{4m^2}{6m} \) in our problem. Here, "numerator" refers to the expression above the division line, which is \( 4m^2 \), and "denominator" is the expression below, \( 6m \).

It's important to:

• Identify these parts correctly, as simplification often requires individually examining both.
• Perform operations like simplification separately on the numerator and the denominator when needed.

By isolating these components, we can apply our knowledge of exponents and factors more effectively. For example, initially, you simplify any numerical constants present in the numerator and denominator, which in this case, would involve reducing the constants 4 and 6.

This breakdown process not only helps to achieve the simplest form of expressions but also aids in understanding the overall structure of algebraic fractions.

Greatest Common Factor

Simplifying fractions involves finding the Greatest Common Factor (GCF) of the numbers involved. The GCF is the largest number that can divide both the numerator and the denominator without leaving a remainder. This is an essential step in reducing fractions.

Let's see how to find the GCF in the problem \( \frac{4m^2}{6m} \):

• First, consider the coefficients in both the numerator and denominator: 4 and 6.
• Identify the GCF of these two numbers, which is the highest number that divides both. Here, it is 2.
• Divide both the numerator and the denominator by this GCF. So 4 divided by 2 equals 2, and 6 divided by 2 equals 3.

This gives us a new, simplified fraction of \( \frac{2m^2}{3m} \), prepared for further simplification through the properties of exponents. Always remember, finding and using the GCF is a powerful tactic to simplify fractions more efficiently and effectively.