Problem 95 - Algebra and Trigonometry

Question

Rationalize the numerator. $$ \frac{1-\sqrt{5}}{3} $$

Step-by-Step Solution

Answer

The rationalized numerator form is \( \frac{-4}{3(1+\sqrt{5})} \).

Step 1: Identify the Problem

We need to rationalize the numerator of the expression \( \frac{1-\sqrt{5}}{3} \). This process involves eliminating the radical (\(\sqrt{5}\)) from the numerator.

Step 2: Find the Conjugate

To rationalize, we multiply both the numerator and the denominator by the conjugate of the numerator, which is \(1+\sqrt{5}\). This helps in eliminating the square root from the numerator.

Step 3: Multiply Numerator and Denominator

Multiply the numerator and denominator by \(1+\sqrt{5}\):\[\frac{(1-\sqrt{5})(1+\sqrt{5})}{3(1+\sqrt{5})}\]

Step 4: Simplify the Numerator

Use the formula for the difference of squares, \((a-b)(a+b) = a^2 - b^2\), to simplify the numerator: \(1^2 - (\sqrt{5})^2 = 1 - 5 = -4\).

Step 5: Simplify the Expression

Substitute back into the fraction to get:\[\frac{-4}{3(1+\sqrt{5})}\] This is the rationalized form of the original expression with respect to the numerator.

Key Concepts

Conjugate

In mathematics, a **conjugate** is a technique used to eliminate square roots (radicals) from the numerator or denominator of a fraction. Conjugates are paired expressions involving the change of a sign between two terms. For example, the conjugate of the expression \(1 - \sqrt{5}\) is \(1 + \sqrt{5}\).

• By multiplying an expression by its conjugate, we aim to cancel out radicals using a special algebraic product called the difference of squares.
• This simplifies the expression because multiplying by the conjugate results in whole numbers.

Multiplying with conjugates is like using a mirror image, where the operation maintains balance by making the denominator or numerator free of square roots.

Difference of squares

The **difference of squares** is a powerful algebraic identity represented by the formula:\[(a - b)(a + b) = a^2 - b^2\]This identity allows you to simplify products of conjugates. Here's how it works:

• In the given exercise, the conjugate \( (1+\sqrt{5}) \) was used to multiply the numerator \( (1-\sqrt{5})(1+\sqrt{5}) \) which forms a difference of squares scenario.
• Applying the formula, \(a = 1\) and \(b = \sqrt{5}\), results in \(1^2 - (\sqrt{5})^2\).
• Calculate each square to get \(1 - 5\), which simplifies directly to \(-4\).

This transformation elegantly eradicates the radical, simplifying the fraction to have non-radical numbers in crucial places.

Simplifying radicals

**Simplifying radicals** is another key concept in algebra that involves removing square roots from expressions when possible. Here are the steps typically involved:

• Identify and multiply by a conjugate to eliminate radicals from the numerator or denominator.
• Calculate using the difference of squares to simplify radical terms.
• Re-write the expression without the radical, ensuring it is simpler and more manageable.

For the example given, after using the conjugate and the difference of squares, the numerator became a simple integer \(-4\), allowing the fraction \(\frac{-4}{3(1+\sqrt{5})}\) to be more streamlined. Radical simplification is common in making expressions easier to work with for further algebraic operations.