Problem 35 - Algebra A Combined Function

Question

Rewrite each rational expression as an equivalent rational expression with the given denominator. $$ \frac{3}{2 x}=\frac{\underline{\phantom{xx}}}{4 x^{2}} $$

Step-by-Step Solution

Answer

The equivalent expression is \( \frac{6x}{4x^2} \).

Step 1: Identify the Given Expression and Target Denominator

You need to rewrite the rational expression \( \frac{3}{2x} \) so that it has the denominator \( 4x^2 \).

Step 2: Analyze the Difference between Denominators

The original denominator is \( 2x \) and the target denominator is \( 4x^2 \). Determine what needs to be multiplied to get from \( 2x \) to \( 4x^2 \).

Step 3: Find the Multiplicative Factor

To transform \( 2x \) into \( 4x^2 \), notice that \( 4x^2 \) is equal to \( 2x \times 2x \). Thus, the multiplicative factor is \( 2x \).

Step 4: Multiply Numerator and Denominator

Multiply both the numerator and the denominator of \( \frac{3}{2x} \) by \( 2x \) to keep the value of the expression unchanged: \( \frac{3 \times 2x}{2x \times 2x} = \frac{6x}{4x^2} \).

Step 5: Confirm the Final Expression

The equivalent rational expression with the denominator \( 4x^2 \) is \( \frac{6x}{4x^2} \).

Key Concepts

Equivalent Expressions

When working with rational expressions, the goal is often to create equivalent expressions. Two expressions are considered equivalent when they have the same value, even though they may look different. For rational expressions, this means that they might have different numerators and denominators but still represent the same quantity.

To create an equivalent expression, you can multiply both the numerator and the denominator by the same non-zero factor. This preserves the value of the original expression because multiplying by the same factor is like multiplying by 1. To understand this concept, consider the rational expression \[ rac{3}{2x} \].

You can form an equivalent expression by identifying a new denominator that you want \[ rac{3}{2x} \] to have and finding the necessary factor to reach it. In our example, the new denominator is \[ 4x^2 \]. To achieve this equivalent expression, you determined that the factor was \[ 2x \]. By multiplying both the numerator \[ 3 \] and the denominator \[2x \] by \[2x \], the equivalent expression you get is \[ rac{6x}{4x^2} \].

So remember, equivalent expressions may appear differently, but their values are always the same!

Denominators

The denominator in a rational expression is the part of the fraction beneath the line. It indicates how many equal parts something is divided into. The denominator plays a crucial role in determining the value of the entire expression.

When you are asked to rewrite a rational expression with a different denominator, it's important to understand both the original and the new denominators. Let's revisit the example featuring the original denominator \(2x\) and the desired denominator \(4x^2\).

The goal is to find a way to transform \(2x\) into \(4x^2\). This usually requires identifying what term is needed in the multiplication process. In our step-by-step solution, the transformation involves multiplying by \(2x\). This is because if you multiply \(2x\) by \(2x\), you get \(4x^2\), matching the desired new denominator.

Understanding and manipulating denominators correctly is essential not only in simplifying expressions but also in solving rational equations. Make sure to always confirm your work by multiplying to ensure the original fraction and its new form are truly equivalent.

Rational Equations

Rational equations are equations that involve fractions whose numerators and denominators are polynomials. Solving rational equations often involves finding a common denominator, which helps to simplify the process of handling fractions.

In our example with rational expressions, while not directly solving an equation, the same principles apply. If you were given an equation such as \( \frac{3}{2x} = \frac{b}{4x^2} \), you would need to find the value of \(b\) by manipulating the fractions just as we did to achieve equivalent expressions.

Steps involved generally include:

• Identifying common denominators to combine or compare fractions.
• Ensuring the operations performed do not change the equivalence by multiplying by denominators on both sides when necessary.
• Simplifying the expressions while keeping both sides of the equation balanced.

It’s key to keep the rational expressions equivalent when solving these equations, as changing their equality results in a different equation altogether. Therefore, the manipulation techniques used to achieve the desired denominators or simplifying the expression are directly applicable in the context of rational equations.