Problem 47 - College Algebra and Calculus: An Applied Approach

Question

Rewrite the expression by rationalizing the denominator. Simplify your answer.\(\frac{3}{\sqrt{5}+\sqrt{6}}\)

Step-by-Step Solution

Answer

The rewritten expression by rationalizing the denominator of the given expression is \(-3\sqrt{5} + 3\sqrt{6}\)

Step 1: Multiplying by conjugate

To start rationalizing the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of the denominator \(\sqrt{5} + \sqrt{6}\) is \(\sqrt{5}-\sqrt{6}\). Hence, \(\frac{3}{\sqrt{5}+\sqrt{6}} \times \frac{\sqrt{5}-\sqrt{6}}{\sqrt{5}-\sqrt{6}}\)

Step 2: Simplifying the fraction

Multiplication result gives us \(\frac{3(\sqrt{5}-\sqrt{6})}{(\sqrt{5} + \sqrt{6})(\sqrt{5}-\sqrt{6})}\). Now, simplify the denominator using the difference of squares, and distribute 3 in the numerator: \(\frac{3\sqrt{5}-3\sqrt{6}}{5-6}= \frac{3\sqrt{5}-3\sqrt{6}}{-1}\)

Step 3: Final adjustment

Finally, simplify by multiplying by -1, and the expression becomes: \(-3\sqrt{5} + 3\sqrt{6}\)

Key Concepts

Difference of Squares

The difference of squares is a useful algebraic identity that helps us simplify expressions, especially when dealing with conjugates. It states that for any two numbers, such as \(a\) and \(b\), the expression \(a^2 - b^2\) can be factored into \((a+b)(a-b)\). This comes in handy when rationalizing denominators, as it allows us to convert an expression with radicals into a simpler one with rational numbers.

In the context of our problem, we have a denominator \((\sqrt{5} + \sqrt{6})(\sqrt{5} - \sqrt{6})\). When applying the difference of squares identity here,