Problem 91 - Intermediate Algebra

Question

Rationalize each denominator. All variables represent positive real numbers. $$ \frac{\sqrt{2}}{\sqrt{5}+3} $$

Step-by-Step Solution

Answer

The rationalized expression is \(-\frac{\sqrt{10}}{4} + \frac{3\sqrt{2}}{4}\).

Step 1: Identify the Conjugate

To rationalize the denominator \( \sqrt{5} + 3 \), we need to multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \( \sqrt{5} + 3 \) is \( \sqrt{5} - 3 \).

Step 2: Multiply Numerator and Denominator by the Conjugate

Multiply \( \frac{\sqrt{2}}{\sqrt{5} + 3} \) by \( \frac{\sqrt{5} - 3}{\sqrt{5} - 3} \). This gives:\[\frac{\sqrt{2}(\sqrt{5} - 3)}{(\sqrt{5} + 3)(\sqrt{5} - 3)}\]

Step 3: Simplify the Numerator

Distribute \( \sqrt{2} \) in the numerator:\[\sqrt{2} \cdot \sqrt{5} - \sqrt{2} \cdot 3 = \sqrt{10} - 3\sqrt{2}\]

Step 4: Simplify the Denominator Using Difference of Squares

Use the difference of squares formula \((a+b)(a-b) = a^2 - b^2\), where \(a = \sqrt{5}\) and \(b = 3\):\[(\sqrt{5})^2 - 3^2 = 5 - 9 = -4\]

Step 5: Write the Rationalized Expression

Now that we have both the numerator and denominator simplified we write:\[\frac{\sqrt{10} - 3\sqrt{2}}{-4}\]Which simplifies to:\[-\frac{\sqrt{10}}{4} + \frac{3\sqrt{2}}{4}\]

Key Concepts

The Power of Conjugates in Rationalizing Denominators

Rationalizing the denominator involves a few straightforward steps, and one of the key components is using a mathematical technique called conjugates. When you hear the term "conjugate," it's simply about changing the sign between two terms.

• Consider the denominator \( \sqrt{5} + 3 \). Its conjugate will be \( \sqrt{5} - 3 \).
• By multiplying both the numerator and the denominator of our fraction by this conjugate, we obtain \( \frac{\sqrt{2}(\sqrt{5} - 3)}{(\sqrt{5} + 3)(\sqrt{5} - 3)} \). This strategic move helps us eliminate the square root in the denominator.

The reason conjugates work so well is because of the difference of squares property, which helps simplify the expression further.

Exploring the Difference of Squares Formula

The difference of squares is a powerful algebraic identity that simplifies expressions when dealing with conjugates. It’s expressed as \( (a + b)(a - b) = a^2 - b^2 \). In the context of rationalizing denominators:

• Given \( a = \sqrt{5} \) and \( b = 3 \), the product \( (\sqrt{5} + 3)(\sqrt{5} - 3) \) simplifies to \((\sqrt{5})^2 - 3^2 \).
• Here, that reduces to \( 5 - 9 = -4 \). This removes the square root, making the expression much tidier.

This principle is fundamental in algebra and is an efficient way to handle terms involving square roots by reducing the complexity of denominators.

Simplifying Numerator and Denominator Expressions

Once the more challenging part of dealing with the conjugate has been tackled, it's time to simplify both the numerator and denominator. Let's break it down:

• Simplifying the numerator involves straightforward distribution: \( \sqrt{2} \cdot \sqrt{5} - \sqrt{2} \cdot 3 \) results in \( \sqrt{10} - 3\sqrt{2} \).
• As for the denominator, thanks to the difference of squares, we already simplified it as \(-4\).

Combining these results gives us the expression \( \frac{\sqrt{10} - 3\sqrt{2}}{-4} \), which further simplifies to \(-\frac{\sqrt{10}}{4} + \frac{3\sqrt{2}}{4} \). By consistently applying mathematical identities and simplification techniques, complex expressions can become manageable.