Problem 12 - Algebra 2 and Trigonometry

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{2 \sqrt{2}}{4 \sqrt{3}}\)

Step-by-Step Solution

Answer

The simplified form is \(\frac{\sqrt{6}}{6}\).

Step 1: Identify the Problem

We need to rationalize the denominator and simplify the expression \(\frac{2 \sqrt{2}}{4 \sqrt{3}}\). Rationalizing the denominator means removing any radical signs from the denominator.

Step 2: Set Up the Multiplication

To remove the square root from the denominator, multiply both the numerator and the denominator by \(\sqrt{3}\). This gives us: \[\frac{2 \sqrt{2}}{4 \sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2 \sqrt{2} \cdot \sqrt{3}}{4 \cdot \sqrt{3} \cdot \sqrt{3}}.\]

Step 3: Simplify the Radicals

Calculate the multiplication in both the numerator and the denominator: \[2 \sqrt{2} \cdot \sqrt{3} = 2 \sqrt{6} \] and \[4 \cdot \sqrt{3} \cdot \sqrt{3} = 4 \cdot 3 = 12.\] So the expression becomes: \[\frac{2 \sqrt{6}}{12}.\]

Step 4: Reduce the Fraction

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2: \[\frac{2 \sqrt{6}}{12} = \frac{\sqrt{6}}{6}.\]

Step 5: Final Expression

Ensure the square root is only present in the numerator and verify if the fraction is simplified. The simplified expression is \(\frac{\sqrt{6}}{6}\).

Key Concepts

Simplify Radicals

Simplifying radicals involves reducing a radical to its simplest form. This means breaking down components under a square root into smaller factors. If any of these factors can be taken out of the radical as a whole number, they should be.

• For example, if you simplify \(\sqrt{18}\), you break it down into \(\sqrt{9 \times 2}\), which results in \(\sqrt{9} \times \sqrt{2} = 3\sqrt{2}\).
• This process makes things easier to work with, especially when there's more arithmetic to do after simplifying.

Be clear, every time you simplify a radical, you're turning it as simple as possible without changing its value.
In our initial exercise, we simplified the expression from \(2\sqrt{2} \cdot \sqrt{3}\) to \(2\sqrt{6}\) by multiplying the numbers inside the radicals and leaving it combined since it's already in the simplest form.

Simplifying Fractions

Simplifying fractions means to reduce the numerical values of the numerator and the denominator by their greatest common divisor.

• When you have \(\frac{6}{12}\), you recognize that both 6 and 12 can be divided by 6, and the fraction simplifies to \(\frac{1}{2}\).
• This ensures that the fraction is as simple as possible, making the math less complex in subsequent steps.

Sometimes, after rationalizing denominators, we end up with fractions that aren't in simplest form. Reducing them ensures clarity and precision.
In the example, after computing \(\frac{2\sqrt{6}}{12}\), we identified that both 2 and 12 share a divisor of 2, which reduced our expression to \(\frac{\sqrt{6}}{6}\). This is the simplest form.

Square Roots

Square roots are numbers that produce a specified number when multiplied by themselves. For example, the square root of 4 is 2, because 2 multiplied by itself equals 4. Square roots are fundamental in expressions that involve radical notation, often seen as \(\sqrt{}\).Square roots often appear in both numerators and denominators, especially in algebraic expressions. They need to be carefully managed during operations such as multiplication or division.

• When multiplying two square roots, like \(\sqrt{a} \cdot \sqrt{b}\), the result can be simplified to \(\sqrt{ab}\).
• This is a result of how the property of radicals operates. It’s important to simplify square roots early on to avoid errors later.

In the given exercise, the rationalizing process utilized this principle, converting \(\sqrt{3} \cdot \sqrt{3}\) in the denominator to 3, thereby creating a new simpler fraction.