Problem 61 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\frac{3 \sqrt{2}}{4 \sqrt{3}}\)

Step-by-Step Solution

Answer

\( \frac{\sqrt{6}}{4} \)

Step 1: Rationalize the Denominator

To eliminate the square root from the denominator, multiply both the numerator and the denominator by \( \sqrt{3} \) which will help simplify the expression.\[ \frac{3 \sqrt{2}}{4 \sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{3 \sqrt{2} \cdot \sqrt{3}}{4 \cdot \sqrt{3} \cdot \sqrt{3}} \]

Step 2: Simplify the Expression

Multiply the expressions in the numerator and the denominator. Notice that \( \sqrt{3} \times \sqrt{3} = 3 \).\[ \frac{3 \sqrt{6}}{4 \cdot 3} = \frac{3 \sqrt{6}}{12} \]

Step 3: Reduce the Fraction

Divide both the numerator and the denominator by their greatest common divisor, which is 3.\[ \frac{3 \sqrt{6}}{12} = \frac{\sqrt{6}}{4} \]

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is a method used to eliminate square roots or cube roots from the denominator of a fraction. This process is important to ensure that the expression is in its simplest form, making it easier to interpret and use in further calculations. To rationalize the denominator, we multiply both the numerator and the denominator of the fraction by the radical that appears in the denominator. This is often where students need to focus, as it requires careful calculation to avoid errors.

In our exercise, we encountered a fraction \( \frac{3 \sqrt{2}}{4 \sqrt{3}} \). To eliminate the \( \sqrt{3} \) from the bottom, we multiply top and bottom by \( \sqrt{3} \), like this:

• Numerator becomes: \( 3 \sqrt{2} \times \sqrt{3} = 3 \sqrt{6} \)
• Denominator becomes: \( 4 \sqrt{3} \times \sqrt{3} = 4 \times 3 = 12 \)

As a result, the fraction simplifies to \( \frac{3 \sqrt{6}}{12} \). This step is crucial for simplifying the radical expression and preparing it for further simplification.

Multiplying Radicals

Multiplying radicals involves combining expressions under a single radical, which requires a firm grasp of square roots and properties of radicals. Here's a straightforward way to approach it: when you multiply two radicals with the same index, you multiply the numbers inside the radical symbols. Remember, \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \). This principle helps in reducing expressions to a simpler form.

In practice, within the given exercise, we multiply the radicals in the numerator: \( \sqrt{2} \times \sqrt{3} = \sqrt{6} \). Therefore, the numerator \( 3 \sqrt{2} \cdot \sqrt{3} \) simplifies to \( 3 \sqrt{6} \). This simplification helps in achieving the goal of the simplest radical form.

Simplifying Fractions

Simplifying fractions is an integral part of dealing with radical expressions. Once the radicals are rationalized and multiplied, the fraction must be simplified to its lowest terms. This ensures clarity and ease when using the expression further.

To simplify a fraction, identify and divide both the numerator and the denominator by their greatest common divisor (GCD). In our example, the expression \( \frac{3 \sqrt{6}}{12} \) can be simplified by noting that both 3 and 12 can be divided by their GCD, which is 3.

• Divide the numerator: \( 3 \div 3 = 1 \)
• Divide the denominator: \( 12 \div 3 = 4 \)

Thus, the simplified expression is \( \frac{\sqrt{6}}{4} \). Simplifying fractions is an essential step in achieving the cleanest and most concise form of a radical expression. This allows for easier interpretation and use of the expression in further math problems.