Problem 46 - College Algebra and Calculus: An Applied Approach

Question

Rewrite the expression by rationalizing the denominator. Simplify your answer.\(\frac{5 x}{\sqrt{14}-2}\)

Step-by-Step Solution

Answer

The simplified form of the expression \(\frac{5 x}{\sqrt{14}-2}\) after rationalizing the denominator is \(\frac{1}{2}x\sqrt{14}+x\).

Step 1: Identify the conjugate

The conjugate of the denominator \(\sqrt{14}-2\) is \(\sqrt{14}+2\). This is derived by changing the sign of the second term in the denominator.

Step 2: Multiply by the conjugate

Multiply the entire expression by \(\frac{\sqrt{14}+2}{\sqrt{14}+2}\), this does not change the value of the expression, since \(\frac{\sqrt{14}+2}{\sqrt{14}+2}\) is actually just 1. This results in the expression \(\frac{5x(\sqrt{14}+2)}{(\sqrt{14}-2)(\sqrt{14}+2)}\).

Step 3: Simplify the denominator

The denominator can be simplified using the difference of two squares formula \(a^2 - b^2 = (a-b)(a+b)\). This results in \(14 - 4 = 10\). So, the expression becomes \(\frac{5x(\sqrt{14}+2)}{10}\).

Step 4: Simplify the numerator

Distribute \(5x\) in \(\sqrt{14}+2\), to get \(5x\sqrt{14}+10x\).

Step 5: Final Simplification

Finally, let's divide both terms in the numerator by the denominator 10 to simplify the expression further: \(\frac{5x\sqrt{14}}{10}+\frac{10x}{10}= \frac{1}{2}x\sqrt{14}+x\).

Key Concepts

Conjugate Multiplication

When dealing with expressions that have a radical in the denominator, you often employ a technique called conjugate multiplication. The conjugate of a binomial like \(\sqrt{14} - 2\) is simply \(\sqrt{14} + 2\). This involves changing the sign between the terms of the radical expression.
Why use the conjugate? Multiplying by the conjugate is crucial because it helps eliminate the square root (or radical) in the denominator. This process leaves you with a simpler, cleaner expression to work with.
Remember:

• The conjugate of \(a - b\) is \(a + b\).
• Always multiply both the numerator and the denominator by this conjugate to maintain the equality of the fraction.

In our example, multiplying by \(\frac{\sqrt{14} + 2}{\sqrt{14} + 2}\) simplifies the denominator drastically.

Difference of Squares

A key technique used in rationalizing denominators is the difference of squares identity. The formula is \(a^2 - b^2 = (a - b)(a + b)\). In simple terms, this formula shows that the product of conjugates will eliminate the radicals and result in a plain number.
In the exercise, we have the conjugates \((\sqrt{14} - 2)\) and \((\sqrt{14} + 2)\). Applying the difference of squares formula:

• \(a = \sqrt{14}\), and \(b = 2\).
• This results in \(a^2 - b^2 = 14 - 4 = 10\).

This step changes the original denominator into a more workable number, 10, significantly simplifying the expression.

Simplifying Expressions

After rationalizing the denominator, you must simplify the entire expression to make it as streamlined as possible. Begin by simplifying the numerator through distribution.
With \(5x\) multiplied by \((\sqrt{14} + 2)\), you carry out a distribution:

• \(5x \cdot \sqrt{14} = 5x\sqrt{14}\)
• \(5x \cdot 2 = 10x\)

This results in \(5x\sqrt{14} + 10x\). Next, simplify this fractional expression\(\frac{5x\sqrt{14}+10x}{10}\) by dividing each component of the numerator by the denominator 10:

• \(\frac{5x\sqrt{14}}{10} = \frac{1}{2}x\sqrt{14}\)
• \(\frac{10x}{10} = x\)

Thus, after simplifying, the expression becomes \(\frac{1}{2}x\sqrt{14} + x\), which is much easier to comprehend and handle.