Problem 7 - Algebra 2 and Trigonometry

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{15}{5 \sqrt{3}}\)

Step-by-Step Solution

Answer

The simplest form of \(\frac{15}{5\sqrt{3}}\) is \(\sqrt{3}\).

Step 1: Identify the Goal

To rationalize the denominator of the fraction \(\frac{15}{5\sqrt{3}}\), we need to remove the square root from the denominator.

Step 2: Multiply by the Conjugate

To eliminate the square root from the denominator, multiply both the numerator and the denominator of the fraction by \(\sqrt{3}\), which is the square root present in the denominator. This will become \(\frac{15 \times \sqrt{3}}{5 \times \sqrt{3} \times \sqrt{3}}\).

Step 3: Simplify the Denominator

Calculate the expression \(5 \times \sqrt{3} \times \sqrt{3}\). Since \(\sqrt{3} \times \sqrt{3} = 3\), the denominator becomes \(5 \times 3 = 15\).

Step 4: Simplify the Numerator

Calculate the expression in the numerator \(15 \times \sqrt{3}\) to get \(15\sqrt{3}\).

Step 5: Simplify the Fraction

Now the fraction is \(\frac{15\sqrt{3}}{15}\). Simplify this by canceling out the common factor \(15\), resulting in \(\sqrt{3}\).

Step 6: Finalize the Expression

The expression is now in its simplest form: \(\sqrt{3}\).

Key Concepts

Simplifying Radicals

Simplifying radicals is an essential skill in mathematics when dealing with square roots and higher roots. The key idea is to express the square root in the simplest possible form. When simplifying a radical, you want to look for squares within the radicand (the number inside the root sign) and simplify them to their base values.

For example, if you have \(\sqrt{12}\), you would factor 12 into its prime factors: \(12 = 2 \times 2 \times 3\). You can then pair up the twos, since \(2 \times 2 = 4\), and we know \(\sqrt{4} = 2\). This allows you to simplify \(\sqrt{12}\) to \(2\sqrt{3}\).

Here’s another tip for simplifying radicals:

• Always look for perfect square factors and simplify them first.
• If the radicand is a product, try to see if you can break it down into smaller, more manageable parts.
• Watch for opportunities to simplify fractions involving radicals by handling each part of the fraction individually.

A simplified radical is easier to work with and understand, especially when resolving more complex equations.

Multiplying by the Conjugate

Multiplying by the conjugate is a technique used to simplify expressions involving radicals, especially when you encounter a binomial with a radical as a denominator. However, when the denominator is a single term radical, such as in our exercise \(\frac{15}{5\sqrt{3}}\), we use something very similar but simpler: multiplication by the "radical over itself."

Multiplying by a radical over itself is essentially using the idea that \(\sqrt{a} \times \sqrt{a} = a\). For our exercise, we utilize \(\sqrt{3} \times \sqrt{3} = 3\). Hence, multiplying \(5\sqrt{3}\) by \(\sqrt{3}\) yields \(15\); thus, removing the square root from the denominator.

• This method effectively "rationalizes" the denominator by removing any irrational numbers.
• It ensures that the entire expression becomes simpler and easier to further simplify.
• Always multiply both numerator and denominator to maintain the equality of the original fraction.

Multiplying by the conjugate or the radical over itself is a strategic move to achieve easier manipulable expressions.

Fraction Simplification

Fraction simplification is the process of reducing fractions to their simplest form, where numerator and denominator have no common factors other than 1. In our task, we started with \(\frac{15}{5\sqrt{3}}\) and aimed to reduce this to its simplest expression.

First, we executed factor cancellations: both the numerator and denominator contain the factor 15 after the rationalizing step. By dividing both by 15, the fraction simplifies neatly to \(\frac{\sqrt{3}}{1}\), which is simply \(\sqrt{3}\).

When simplifying fractions:

• Always look for common factors in the numerator and denominator.
• Divide both terms by their greatest common divisor (GCD) for the simplest form.
• Ensuring fractions are simplified makes solving subsequent problems easier.

A simplified fraction not only looks cleaner but represents the true relationship between the numerator and denominator in comparison to each other. It's an essential step in rationalizing and refining mathematical solutions.