91Ó°ÊÓ

When you encounter a fraction with a denominator that contains a square root, you can "rationalize" the denominator to make it simpler. Rationalizing involves eliminating the radical from the denominator. The key tool here is the "conjugate pair." The conjugate of a binomial like \(a + b\) is \(a - b\), and vice versa. The beauty of using conjugates lies in their ability to eliminate radicals when multiplied together. For instance, if the denominator is \(\sqrt{x} + 3\), its conjugate is \(\sqrt{x} - 3\). By multiplying both the numerator and the denominator by this conjugate, the denominator transforms into a more manageable form without radicals. This step is foundational in algebra, particularly in operations involving limits and integrals in calculus.

• Multiplying by a conjugate simplifies the expression by using subtraction.
• Conjugate pairs help in making expressions cleaner and easier to handle.

The difference of squares formula is a powerful algebraic identity used to simplify expressions involving squares. It states that \((a+b)(a-b) = a^2 - b^2\). This property is particularly useful when multiplying a number by its conjugate. As seen in the solution, if we let \(a = \sqrt{x}\) and \(b = 3\), then the expression \((\sqrt{x} + 3)(\sqrt{x} - 3)\) simplifies to \((\sqrt{x})^2 - 3^2\).
With this, we achieve a denominator of \(x - 9\) instead of a complex expression with a square root. The application of this formula is crucial in simplifying not only fractions but also in solving polynomial equations and even factorization in algebra.

• Using the formula quickly simplifies the denominator.
• This approach helps remove square roots efficiently.

The distributive property is one of the basic algebraic properties which states that \(a(b+c) = ab + ac\). This property allows you to multiply a single term by every term inside a set of parentheses. In rationalizing the given expression, this property is used to expand the numerator. With \(\frac{\sqrt{x}(\sqrt{x} - 3)}{x - 9}\), applying the distributive property results in \(x - 3\sqrt{x}\) as we expand the numerator: \(\sqrt{x} \cdot \sqrt{x} = x\) and \(\sqrt{x} \cdot (-3) = -3\sqrt{x}\).
By expanding the expression using the distributive property, we achieve a simplified form, making further calculations, such as integration or solving equations involving the fraction, more straightforward.

• Distributive property simplifies multiplication over addition.
• Helps in breaking down expressions to a simpler form.