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An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators such as addition, subtraction, multiplication, and division. For example, the expression \(\frac{(2 y)^{3}}{15 x}\) is an algebraic expression. In this expression, \(2y\) is raised to the third power. When we simplify this, we cube both 2 and \(y\) to get \(8y^3\). The use of exponents is common in algebraic expressions to represent repeated multiplication. Algebraic expressions can be as simple as a single term or as complex as a fraction within a fraction, known as a complex fraction, which our main problem illustrates.

The importance of understanding algebraic expressions lies in the ability to manipulate and re-write them in different forms — often aiming to simplify the expression to make it easier to understand or to solve an equation. Simplification may involve combining like terms, factoring, expanding, and reducing fractions, which is why it's a staple aspect in algebra courses.

Simplifying fractions is the process of reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1. To simplify a fraction, you divide both the top number (numerator) and bottom number (denominator) by their greatest common divisor.

For example, consider the fraction obtained after applying the reciprocal in the original problem (Step 3): \(\frac{72 y x^{2}}{330 x y^{2}}\). To simplify this, we look for any common factors in the numerator and denominator. Since both numerator and denominator are divisible by 6, we can reduce the fraction, leading us to \(\frac{12 x}{55 y}\) as the simplest form. It is essential to always look for common terms in both the numerator and denominator to simplify fractions effectively. This not only makes the expressions easier to work with but also is crucial in solving algebraic equations.

The reciprocal of a fraction is created by swapping its numerator and denominator. Essentially, for a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are non-zero, the reciprocal is \(\frac{b}{a}\). It is also equivalent to \(1\) divided by the fraction. Taking the reciprocal is particularly useful when dealing with divisions of fractions, as seen in Step 3, where the reciprocal of the bottom fraction is used. The operation changes the original division into multiplication, which simplifies the complex fraction.

Using the reciprocal is an important concept when simplifying complex fractions. It is also useful when solving equations involving fractions and when working with ratios and proportions. Understanding the concept of reciprocals allows students to handle division between fractions more comfortably and is pivotal in advancing algebraic proficiency.