Problem 21 - Elementary Algebra

Question

Simplify each complex rational expression. $$ \frac{4-\frac{1}{m^{2}}}{2+\frac{1}{m}} $$

Step-by-Step Solution

Answer

Question: Simplify the complex rational expression: $$\frac{4-\frac{1}{m^{2}}}{2+\frac{1}{m}}$$ Answer: The simplified complex rational expression is: $$\frac{4m^2 - 1}{2m^3 + m^2}$$

Step 1: Identify the denominators in the complex fractions

In the given expression, the denominators are \(m^2\) and \(m\). Now we'll find the LCM of these denominators.

Step 2: Find the Least Common Multiple (LCM) of the denominators

The given denominators are \(m^2\) and \(m\). We can factor \(m^2\) into \(m\cdot m\) and \(m\) as \(m\). The LCM must contain enough factors to account for each factor in the given denominators. So, the LCM of these denominators is \(m^2\).

Step 3: Multiply the numerator and denominator by the LCM

Multiply both the numerator and denominator of the expression by the LCM of the denominators, which is \(m^2\). This should cancel out the fractions within the expression: $$ \frac{4-\frac{1}{m^{2}}}{2+\frac{1}{m}}\cdot\frac{m^2}{m^2} = \frac{(4-\frac{1}{m^2})\cdot m^2}{(2+\frac{1}{m})\cdot m^2} $$

Step 4: Perform the multiplication in the numerator and denominator

Multiply the terms in the numerator and denominator: Numerator: \((4-\frac{1}{m^2})\cdot m^2 = 4m^2 - 1\) Denominator: \((2+\frac{1}{m})\cdot m^2 = 2m^3 + m^2\) The expression now becomes: $$ \frac{4m^2 - 1}{2m^3 + m^2} $$

Step 5: Simplify the expression

It doesn't seem that we can simplify the expression any further. Therefore, the simplified complex rational expression is: $$ \frac{4m^2 - 1}{2m^3 + m^2} $$

Key Concepts

Least Common Multiple (LCM)

To simplify complex rational expressions, identifying the Least Common Multiple (LCM) is vital. In general, the LCM is the smallest multiple that is evenly divisible by each of the denominators involved. Let's say you have two denominators, like in our example where the denominators are \(m^2\) and \(m\). To find the LCM, factor each term. We have \(m^2\) which can be broken down into \(m \cdot m\) and \(m\) as just \(m\). For the LCM to work, it must include the maximum number of each unique factor from these factorizations.In this scenario, \(m^2\) contains more factors of \(m\). Hence, the LCM of \(m^2\) and \(m\) is \(m^2\). This helps in eliminating the fractional parts when multiplying through the expression.

Simplification

Simplification is a key step when dealing with complex rational expressions. The goal is to rewrite the expression in its simplest form, making it easier to work with or understand. After identifying the LCM, we use it to clear out fractions within the expression. In this process, multiply both the numerator and the denominator by the LCM. For example, using \(m^2\) as the LCM, we multiply to eliminate fractions in the given complex expression: \(\frac{4-\frac{1}{m^{2}}}{2+\frac{1}{m}}\cdot\frac{m^2}{m^2} \)becomes \(\frac{(4-\frac{1}{m^2})\cdot m^2}{(2+\frac{1}{m})\cdot m^2}\).By performing this multiplication, you transform the numerator and the denominator into more straightforward expressions which can often be simplified further, if common factors exist.

Numerator and Denominator Operations

When simplifying complex rational expressions, operations on the numerator and denominator are crucial. Once you've multiplied by the LCM, you carry out multiplication in both parts. This makes your expression more manageable.For the numerator, multiply each term by the LCM: - Original: \(4-\frac{1}{m^2}\)- Multiplied: \((4-\frac{1}{m^2})\cdot m^2 = 4m^2 - 1\)Similarly, do the same for the denominator:- Original: \(2+\frac{1}{m}\)- Multiplied: \((2+\frac{1}{m})\cdot m^2 = 2m^3 + m^2\)The result, \(\frac{4m^2 - 1}{2m^3 + m^2}\), is a simplified form of your expression. Understanding these operations enables you to confidently simplify and manipulate complex rational fractions.