Question

Simplify the expression. $$\frac{4+\sqrt{3}}{a-\sqrt{b}}$$

Short answer

The simplified form of the given expression is \[\frac{4a + 4\sqrt{b} + a\sqrt{3} + \sqrt{3b}}{a^2 - b}\]

Step-by-step solution

Identify the denominator

The denominator of the given fraction \(\frac{4+\sqrt{3}}{a-\sqrt{b}}\) is \(a - \sqrt{b}\). Our goal is to rationalize this denominator, i.e., remove the square root term from the denominator.

Rationalize the denominator

To rationalize the denominator \(a - \sqrt{b}\), multiply both the numerator and denominator of the fraction by the conjugate of the denominator. The conjugate of \(a - \sqrt{b}\) is \(a + \sqrt{b}\). This gives us \[\frac{4+\sqrt{3}}{a-\sqrt{b}} \times \frac{a + \sqrt{b}}{a + \sqrt{b}}\]

Apply the difference of squares formula

The difference of squares formula tells us that \((a-b)(a+b)=a^2 - b^2\). We can apply this in the denominator to get \(a^2 - b\). It should be noted that \sqrt{b} squared gives b. Also, in the numerator, distribute to get \(4a + 4\sqrt{b} + a\sqrt{3} + \sqrt{3b}\).

Write the final expression

The numerator and the denominator obtained in the last step are simplified. The final simplified expression is: \[\frac{4a + 4\sqrt{b} + a\sqrt{3} + \sqrt{3b}}{a^2 - b}\]

Key concepts

Simplify Expressions

Simplifying expressions is a fundamental skill in algebra that involves reducing an algebraic expression to its simplest form. This process often makes it easier to understand and work with the given expression. In our exercise, simplification begins with rationalizing the denominator, converting the complex fraction into one without radicals in the denominator.

Simplification can continue with combining like terms and performing arithmetic operations. For instance, once the denominator of our exercise is rationalized, we might find that the numerator can be simplified further by combining terms that share the same radical or integer part. However, in the provided step-by-step solution, after rationalization, no further simplification is necessary which often occurs when simplifying expressions involving square roots.

Conjugate

The concept of the conjugate is critical when working with complex fractions, particularly when you encounter a radical in the denominator. A conjugate in algebra refers to a binomial formed by changing the sign between two terms. In the context of our exercise, to rationalize the denominator, we use the conjugate of the denominator, which is \(a + \sqrt{b}\).

Why use the conjugate? Multiplying by the conjugate is a useful trick because it takes advantage of the difference of squares property, which effectively eliminates the square root without changing the value of the expression. In the step-by-step solution, multiplying the original fraction by \(1\) in the form of \(\frac{a + \sqrt{b}}{a + \sqrt{b}}\) is the key action that starts the simplification process.

Difference of Squares

The difference of squares is an important factoring technique and plays a central role in rationalizing denominators with radicals. It states that for any two terms \(a\) and \(b\), \(a^2 - b^2 = (a-b)(a+b)\).

When we multiply the conjugate pairs in the rationalization process, we effectively create a difference of squares situation: \(a - \sqrt{b}\) times \(a + \sqrt{b}\) yields \(a^2 - (\sqrt{b})^2\), which simplifies to \(a^2 - b\), a rational number. In the exercise, this property eliminates the radical from the denominator, thus achieving our goal to rationalize the denominator.

Understanding the difference of squares allows us to work effectively with quadratic expressions and to perform operations like rationalization with ease and accuracy.