Question

Simplify the expression. $$\frac{3 x}{8 x^{2}} \cdot \frac{4 x^{3}}{3 x^{4}}$$

Short answer

The simplified expression is \( \frac{1}{2x^{2}} \)

Step-by-step solution

Multiplication of Fraction's Numerators and Denominators

Firstly, start by multiplying the numerators together, and then the denominators together: \( \frac{3x \times 4x^{3}}{8x^{2} \times 3x^{4}} \)

Simplifying Terms

This results in \( \frac{12x^{4}}{24x^{6}} \) which simplifies to \( \frac{x^{4}}{2x^{6}} \)

Reduce the Fraction

Reducing the fraction even further by cancelling \(x^{4}\) in the numerator and the denominator results in \( \frac{1}{2x^{2}} \)

Key concepts

Fraction Multiplication

Understanding how to multiply fractions is fundamental in algebra, especially when dealing with algebraic expressions that contain variables. The process is straightforward: multiply the numerators (the top numbers) with each other and then the denominators (the bottom numbers) with each other.

For instance, in the example given, we have two fractions: \( \frac{3x}{8x^2} \) and \( \frac{4x^3}{3x^4} \). To multiply these, we combine the numerators to get \(3x \times 4x^3\) and the denominators to get \(8x^2 \times 3x^4\). The operation results in \(\frac{12x^4}{24x^6}\), which forms the basis for further simplification.

When multiplying fractions with variables, it is important to remember to apply the same rules as for numerical fractions but also to keep the variable parts consistent with the laws of exponents.

Variable Simplification

The process of variable simplification involves reducing algebraic expressions to their simplest form, which makes them easier to understand and work with. Variables represent unknown quantities and obey particular rules when subjected to mathematical operations.

In the case of our exercise, after multiplying the fractions, we obtain \( \frac{12x^4}{24x^6} \). Now, it's time to simplify this expression. This is where knowledge of exponents is crucial. For example, \(x^4\) divided by \(x^6\) results in \(x^{-2}\), according to the laws of exponents that state when dividing like bases, subtract the exponents. This simplifies to \(\frac{1}{x^2}\) because a negative exponent indicates the reciprocal of that base raised to the positive exponent. This step is vital in reducing the complexity of algebraic fractions to their simplest form, leading to the final simplified result \(\frac{1}{2x^2}\).

Algebraic Fractions

Algebraic fractions are fractions that contain variables, such as \( \frac{3x}{8x^2} \) in our exercise. Working with these means that you'll be applying the rules of arithmetic fractions while also considering the properties of variables and exponents. Simplifying algebraic fractions makes them more manageable and easier to integrate into larger algebraic problems.

Our original exercise showcases a situation where we're able to cancel out like terms in the numerator and denominator to simplify the algebraic fraction. After the multiplication and initial simplification, we reach \( \frac{x^4}{2x^6} \). Here, we can see that \(x^4\) is common to both the numerator and the denominator, allowing us to divide each by \(x^4\). This gives us the final and simpler expression \( \frac{1}{2x^2} \). This step is what often confuses students, but recognizing common factors and understanding exponents can clarify this process significantly, making algebraic fractions less intimidating.