Problem 62 - Algebra and Trigonometry

Question

Rationalize the denominator. (a) \(\frac{12}{\sqrt{3}}\) (b) \(\frac{5}{\sqrt{2}}\) (c) \(\frac{2}{\sqrt{6}}\)

Step-by-Step Solution

Answer

(a) \(4\sqrt{3}\), (b) \(\frac{5\sqrt{2}}{2}\), (c) \(\frac{\sqrt{6}}{3}\).

Step 1: Rationalizing the Denominator for Part (a)

To rationalize the denominator of \( \frac{12}{\sqrt{3}} \), multiply both the numerator and the denominator by \( \sqrt{3} \) to eliminate the square root in the denominator. This gives: \[ \frac{12}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{12\sqrt{3}}{3}. \] Simplify this expression: \[ \frac{12\sqrt{3}}{3} = 4\sqrt{3}. \] Hence, \( \frac{12}{\sqrt{3}} = 4\sqrt{3}. \)

Step 2: Rationalizing the Denominator for Part (b)

To handle \( \frac{5}{\sqrt{2}} \), multiply both numerator and denominator by \( \sqrt{2} \): \[ \frac{5}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{5\sqrt{2}}{2}. \] The rationalized form is \( \frac{5\sqrt{2}}{2} \).

Step 3: Rationalizing the Denominator for Part (c)

For \( \frac{2}{\sqrt{6}} \), multiply the numerator and the denominator by \( \sqrt{6} \): \[ \frac{2}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{2\sqrt{6}}{6}. \] Simplify the fraction \( \frac{2\sqrt{6}}{6} = \frac{\sqrt{6}}{3}. \) Thus, \( \frac{2}{\sqrt{6}} = \frac{\sqrt{6}}{3}. \)

Key Concepts

Simplifying Radical Expressions

Simplifying radical expressions is a crucial skill in algebra that involves breaking down expressions that contain square roots into their simplest form. This makes calculations easier and expressions neater. A radical expression typically involves a root, such as a square root, and simplifying it might mean making the expression free of fractions within the radical.

• Understanding how to identify like radicands (the number inside the square root) is the first step.
• You can simplify a radical expression by factoring out squares from under the root sign. For example, the expression \( \sqrt{18} \) can be simplified by recognizing that \( 18 = 9 \, \times \, 2 \), which leads to \( \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \).
• Always make sure that the expression inside the root is as small as possible.

This helps when simplifying the entire expression and is particularly useful when dealing with fractions. By breaking the expressions down, you can more easily rationalize or further simplify them.

Square Roots

Square roots are a fundamental aspect of algebra, appearing frequently in both simple and complex mathematical problems. A square root of a number \( \sqrt{x} \) is a value that, when multiplied by itself, gives \( x \).

• Understanding basic properties of square roots helps, such as \( \sqrt{x^2} = x \) where \( x \) is a non-negative number.
• Square roots of positive numbers always result in two values: a positive root and a negative root, e.g., \( \sqrt{16} = 4 \) and \( -4 \).
• In operations, particularly with fractions and rationalizing, focusing on the positive root is important for standard solutions.

Being comfortable with square roots enables students to work through simplifications and other transformations more fluently. It's about recognizing patterns and repeatedly applying the concept to ease up mathematical manipulations.

Fractions

Fractions are common in mathematical expressions and represent parts of a whole. In the context of rationalizing denominators, understanding fractions is crucial.

• Fractions have two parts: a numerator (top number) and a denominator (bottom number).
• When dealing with radicals in denominators, the aim is to make the denominator rational—meaning free from square roots or other radicals.
• Rationalizing involves multiplying both the numerator and the denominator by the radical present in the denominator.

For instance, with a fraction like \( \frac{5}{\sqrt{2}} \), multiplying both parts of the fraction by \( \sqrt{2} \) eliminates the radical in the denominator, resulting in \( \frac{5\sqrt{2}}{2} \). This allows for the expression to be considered "rational." Understanding this concept helps in solving a wide variety of mathematical equations more easily and leads to cleaner, more standardized forms.