Question

Find the product. $$ \frac{4 x y^3}{x^2 y} \cdot \frac{y}{8 x} $$

Short answer

The product of the two fractions is \(\frac{y^3}{2 x}\).

Step-by-step solution

Simplify each fraction

First, simplify each fraction to its smallest form. A good way to do this is by cancelling out similar terms from the numerator and the denominator. For the first fraction \(\frac{4 x y^3}{x^2 y}\), \(x\) and \(y\) can be cancelled from the numerator and the denominator. This will leave \( \frac{4 y^2}{x}\). Similarly for the second fraction \(\frac{y}{8 x}\), no simplification is possible.

Multiply the simplified fractions

Now that the fractions have been simplified, multiply them. This involves multiplying the numerators with each other and the denominators with each other. So, \(\frac{4 y^2}{x} \cdot \frac{y}{8 x} = \frac{4 y^2 \cdot y}{x \cdot 8 x}\)

Simplify resulting product

Finally, simplify the product by cancelling out any common factors from the numerator and the denominator. The \(x\) in the denominator can be cancelled with one of the \(x\)s in the denominator. And number 4 in the numerator can be cancelled with the 8 in the denominator to give 2. This will simplify the fraction to \(\frac{y^3}{2 x}\)

Key concepts

Simplifying Algebraic Fractions

Getting comfortable with simplifying algebraic fractions is crucial in mastering algebra. The idea is to reduce the expression to its most basic form, where no further reduction is possible. Start by looking for common factors in the numerator and the denominator that can be divided out. For example, consider an algebraic fraction like \(\frac{4 x y^3}{x^2 y}\). Here, you can cancel out an \(x\) because it's present in both the numerator and the denominator. Likewise, \(y\) can also be reduced. After cancelling out these terms, you're left with \(\frac{4 y^2}{x}\), a much simpler expression.

It's a good habit to first simplify before moving on to any other operations with fractions. This not only makes your work neater but can also simplify the arithmetic you have to deal with going forward. The goal is to minimize the complexity of subsequent steps, such as multiplication or addition of fractions.

Multiplication of Algebraic Expressions

When working with multiplication of algebraic expressions, it is quite analogous to multiplying numerical fractions—multiply the numerators together, and the denominators together. Say you have two simplified algebraic fractions, such as \(\frac{4 y^2}{x}\) and \(\frac{y}{8x}\). To find their product, you multiply the numerators (\(4y^2\) and \(y\)) to get \(4y^3\), and the denominators (\(x\) and \(8x\)) to get \(8x^2\). This gives you the fraction \(\frac{4 y^3}{8 x^2}\).

Multiplying algebraic expressions correctly is vital in many areas of mathematics and can be applied to solve equations and understand other complex functions. Keep an eye on simplifying at each step, as it can make the multiplication process smoother and the result clearer.

Rational Expressions

A rational expression is much like a fraction, but instead of integers, you have polynomials in the numerator and denominator. Behaving under the same principles as regular fractions, rational expressions can also be added, subtracted, multiplied, and divided. The primary goal with rational expressions, however, is to simplify. In the context of our exercise with the expression \(\frac{y^3}{2 x^2}\), after multiplication and simplification, we observe that unless all common factors are canceled, we do not achieve the simplest form of the rational expression.

To simplify further, always look for common polynomial factors that can be divided out. For students, understanding that these algebraic fractions can be manipulated in the same way as numerical ones is an important stepping stone in algebra.