Problem 17 - Intermediate Algebra

Question

Rationalize each denominator. See Examples 1 through 3. $$ \frac{2 \sqrt{3}}{\sqrt{7}} $$

Step-by-Step Solution

Answer

The rationalized form is \( \frac{2 \sqrt{21}}{7} \).

Step 1: Understand the Problem

Rationalizing a denominator involves removing a square root from the denominator of a fraction. In this problem, we have \( \frac{2 \sqrt{3}}{\sqrt{7}} \) where the denominator is \( \sqrt{7} \). We need to eliminate this irrational number from the denominator.

Step 2: Multiply by the Conjugate

To rationalize the denominator, multiply both the numerator and the denominator by \( \sqrt{7} \). This is because the product of \( \sqrt{7} \) times itself results in a rational number.\[ \frac{2 \sqrt{3}}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{2 \sqrt{3} \cdot \sqrt{7}}{\sqrt{7} \cdot \sqrt{7}} \]

Step 3: Simplify Numerator and Denominator

Calculate the product in the numerator and the denominator separately:- The numerator becomes: \( 2 \sqrt{3} \cdot \sqrt{7} = 2 \sqrt{21} \)- The denominator becomes: \( \sqrt{7} \cdot \sqrt{7} = 7 \)

Step 4: Write the Rationalized Expression

The rationalized expression is:\[ \frac{2 \sqrt{21}}{7} \] The denominator is now rational.

Key Concepts

Radicals

Radicals are symbols used to represent the root of a number. Most commonly, they are used for square roots, which is noted with the radical symbol \( \sqrt{} \). This symbol allows us to visually express when a number is raised to the power of 1/2, hence finding its root.

• For example, \( \sqrt{9} = 3 \) because 3 squared is 9.
• Radicals are also used for cube roots \( \sqrt[3]{} \), fourth roots, and so on.

When working with fractions, like in the original exercise \( \frac{2 \sqrt{3}}{\sqrt{7}} \), having a radical in the denominator presents a unique challenge. By using radicals in the denominator, we face potential difficulties in calculations and approximations, motivating the need to rationalize it.
In essence, the operation of rationalizing involves neutralizing the radical sign within a denominator, transforming the expression into an equivalent form without a radical in the denominator. The eliminated radical then enables easier mathematical operations with more predictable outcomes.

Irrational Numbers

Irrational numbers are real numbers that cannot be expressed as a simple fraction of two integers. They are the numbers which have non-repeating and non-terminating decimal expansions.

• Common examples include \( \pi \), \( e \) (Euler’s number), and square roots of prime numbers like \( \sqrt{2} \), \( \sqrt{3} \), and \( \sqrt{7} \).

In the context of radicals and rationalization, numbers under a square root that don't result in a perfect square often remain irrational. For instance, \( \sqrt{7} \) in the original exercise cannot be further simplified to a clean integer.
When irrational numbers appear in the denominator of a fraction, it's customary to rationalize the expression. Doing so helps simplify computations and allows for clearer results when the number is part of further operations or equations.

Multiplying by Conjugates

Conjugates are pairs of expressions that can be used to eliminate irrational numbers from a denominator. It's a technique used intentionally to simplify algebraic expressions. In this method, one multiplies an expression by a form of 1 that contains the conjugate of the denominator.

• The conjugate of a single square root \( \sqrt{7} \) is just itself \( \sqrt{7} \), used as seen in the original exercise.
• For complex expressions involving sums or differences, like \( a + \/sqrt{b} \), the conjugate is \( a - \/sqrt{b} \).

When you multiply a fraction by its conjugate form, both the numerator and the denominator must be multiplied by this conjugate to ensure the result is equivalent to the original fraction. For example, multiplying \( \frac{2 \sqrt{3}}{\sqrt{7}} \) by \( \sqrt{7}/\sqrt{7} \) rationalizes the denominator because \( \sqrt{7} \times \sqrt{7} = 7 \) becomes a rational number.
Using conjugates is a powerful algebraic method that allows for efficient simplification of expressions, enabling easier mathematical operations and clearer interpretations of the results.