Solution: Multiply numerator and denominator by the conjugate of the denominator, $\sqrt{7} - 3$:

["Title: Mastering Rationalizing the Denominator: Multiply Numerator and Denominator by the Conjugate", "Meta Description:\nLearn how to rationalize the denominator of a fraction by multiplying both numerator and denominator by the conjugate—specifically $\sqrt{7} - 3$. This proven algebra technique simplifies complex expressions and improves mathematical clarity.", "---", "When solving algebra problems involving fractions, one common challenge is simplifying expressions with irrational denominators. A powerful and widely used method to eliminate square roots from denominators is multiplying both the numerator and denominator by the conjugate of the denominator. In this article, we focus on the technique of multiplying by $\sqrt{7} - 3$ to rationalize denominators, especially when the denominator contains a square root term like $\sqrt{7}$.", "### Why Rationalize the Denominator?", "Rationalizing the denominator transforms an expression like:", "$$\n\frac{a}{\sqrt{7} - 3}\n$$", "into a cleaner, more usable form:", "$$\n\frac{a(\sqrt{7} + 3)}{(\sqrt{7} - 3)(\sqrt{7} + 3)} = \frac{a(\sqrt{7} + 3)}{7 - 9} = \frac{a(\sqrt{7} + 3)}{-2}\n$$", "This process removes irrational numbers from the denominator, making the expression easier to work with in further calculations, comparisons, or graphing.", "---", "### What is a Conjugate?", "The conjugate of a binomial expression $a - b$ is $a + b$. For denominators involving square roots such as $\sqrt{7} - 3$, multiplying by its conjugate $\sqrt{7} + 3$ is the key step in rationalization. When applied, the product of conjugates uses the difference of squares identity:", "$$\n(x - y)(x + y) = x^2 - y^2\n$$", "So in our example:", "$$\n(\sqrt{7} - 3)(\sqrt{7} + 3) = (\sqrt{7})^2 - (3)^2 = 7 - 9 = -2\n$$", "This simple yet effective transformation eliminates the radical from the denominator.", "---", "### Step-by-Step: Rationalizing $\dfrac{a}{\sqrt{7} - 3}$", "Let’s walk through applying the conjugate method to a generic numerator $a$:", "1. Identify the conjugate:\n For the denominator $\sqrt{7} - 3$, the conjugate is $\sqrt{7} + 3$.", "2. Multiply numerator and denominator by the conjugate:\n $$\n \frac{a}{\sqrt{7} - 3} \cdot \frac{\sqrt{7} + 3}{\sqrt{7} + 3} = \frac{a(\sqrt{7} + 3)}{(\sqrt{7} - 3)(\sqrt{7} + 3)}\n $$", "3. Simplify the denominator using the difference of squares:\n $$\n (\sqrt{7} - 3)(\sqrt{7} + 3) = 7 - 9 = -2\n $$", "4. Write the final simplified expression:\n $$\n \frac{a(\sqrt{7} + 3)}{-2} = -\frac{a(\sqrt{7} + 3)}{2}\n $$", "Voilà! The irrational denominator is gone, and the expression is rationalized.", "---", "### Practical Applications", "This technique is invaluable in high-school algebra, calculus, and advanced math when dealing with irrational denominators. It’s essential in fields such as engineering, physics, and economics where precise, simplified forms are necessary for modeling and analysis.", "---", "### Final Thoughts", "Rationalizing the denominator by multiplying numerator and denominator by the conjugate—like $\sqrt{7} - 3$—is a fundamental skill that boosts algebraic fluency and problem-solving confidence. Once mastered, this method becomes intuitive, enabling smooth manipulation of complex rational expressions.", "Keep practicing—parallel rationalization using conjugates is not just a formula, it’s a powerful tool!", "---", "Keywords: rationalize denominator, multiply by conjugate, rationalize $\sqrt{7} - 3$, algebra technique, conjugate multiplication, simplify irrational denominator, step-by-step example, how to rationalize a fraction denominator, algebra homework help."]