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In mathematics, the concept of a conjugate is quite useful when dealing with expressions involving square roots. For a binomial expression that includes a square root, such as \(a + b\), the conjugate is \(a - b\) or the opposite sign between the terms. This idea helps because multiplying a binomial by its conjugate eliminates the middle term and leaves a simpler algebraic expression.

Consider the denominator in the problem \(\sqrt{5} - 2\). The conjugate of this expression is \(\sqrt{5} + 2\). When we multiply the denominator by its conjugate, it results in an expression that no longer has a square root. This manipulation is a key technique in rationalizing denominators, allowing us to convert the denominator into a rational number. Rationalizing the denominator is important because it simplifies the expression and makes it easier to understand and work with in further calculations.

The difference of squares is a special algebraic theorem that signifies the difference between two square terms. The formula is expressed as:

• \((a^2 - b^2) = (a - b)(a + b)\)

Using this theorem, when multiplying the original binomial \((\sqrt{5} - 2)\) by its conjugate \((\sqrt{5} + 2)\), the outcome is a simplified expression \(\sqrt{5}^2 - 2^2\).

This results in:

• \((\sqrt{5})^2 - (2)^2 = 5 - 4\)

which equals 1. Recognizing and applying the difference of squares formula is essential in achieving simplification during the rationalization process. It conveniently reduces expressions with radicals, transforming them into a more manageable form, which aids in both clarity and computational efficiency.

Simplifying fractions is a crucial step in many mathematical operations, including rationalizing denominators. Once the expression involving the conjugate is expanded, it often results in a more complex numerator and a relatively simple denominator due to the difference of squares. The goal in simplifying fractions is to reduce both the numerator and the denominator to their simplest forms.

In our specific example, after multiplying by the conjugate, the fraction becomes \(\frac{7(\sqrt{5} + 2)}{1}\). We break down the multiplication in the numerator to get \(7\sqrt{5} + 14\). Since the denominator simplifies to 1, the entire process culminates in a straightforward expression \(7\sqrt{5} + 14\). By simplifying the fraction, we remove the radical from the denominator entirely, reaching an expression that is easier to interpret and use in subsequent calculations.