Problem 55 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rationalize the denominator of the expression. $$ \frac{3}{2 \sqrt{x}} $$

Step-by-Step Solution

Answer

The short answer is: To rationalize the denominator of the expression \(\frac{3}{2\sqrt{x}}\), multiply both the numerator and the denominator by the conjugate of the denominator, which is also \(2\sqrt{x}\). After multiplying and simplifying the expression, the rationalized denominator form is \(\frac{3\sqrt{x}}{2x}\).

Step 1: Identify the conjugate of the denominator

The conjugate of the denominator \(2\sqrt{x}\) is the same expression but with an opposite sign for the square root term, which is also \(2\sqrt{x}\).

Step 2: Multiply numerator and denominator by the conjugate

Multiply both the numerator and the denominator by \(2\sqrt{x}\): \(\frac{3}{2\sqrt{x}} \times \frac{2\sqrt{x}}{2\sqrt{x}}\)

Step 3: Simplify the expression

Multiply the numerators and the denominators: \(\frac{3(2\sqrt{x})}{(2\sqrt{x})(2\sqrt{x})}\) Simplify the expression: \(\frac{6\sqrt{x}}{4x}\)

Step 4: Reduce the fraction

We can simplify this expression by factoring out a common factor of 2: \(\frac{6\sqrt{x}}{4x} = \frac{2(3\sqrt{x})}{2(2x)}\) Dividing both the numerator and the denominator by the common factor 2: \(\frac{3\sqrt{x}}{2x}\) The given expression with a rationalized denominator is \(\frac{3\sqrt{x}}{2x}\).

Key Concepts

Conjugate of a Denominator

Understanding the conjugate of a denominator is crucial when rationalizing denominators in an expression. The conjugate of a denominator involves changing the sign between two terms in a binomial. For example, if you have a denominator of the form \( a + b \sqrt{c} \) where \( a \) and \( b \) are numbers and \( c \) is the radicand, the conjugate is \( a - b \sqrt{c} \).

The primary reason for using the conjugate is to eliminate radicals from the denominator of a fraction, creating a rational denominator. When you multiply by the conjugate, the result in the denominator is a difference of squares which simplifies to a non-radical expression. This technique is particularly helpful for rendering complex expressions into a more straightforward and workable form. In the provided exercise, the denominator doesn't have a binomial form but is simply \(2 \sqrt{x}\). Amazingly, the same principle applies, and multiplying by \(2 \sqrt{x}\) again, we effectively use its own conjugate to rationalize it.

Simplify Expressions

Simplifying expressions is a fundamental skill in algebra that makes equations and functions easier to evaluate and comprehend. When we simplify expressions, especially those involving radicals, our goal is to write them in their simplest form without changing their values.

Here are some general steps to simplify expressions:

• Combine like terms (add or subtract variables and numbers that are the same).
• Reduce fractions by dividing the numerator and denominator by their greatest common factor.
• Expand expressions to eliminate parentheses where beneficial.
• Factor expressions to reveal and cancel common factors.

When confronted with a radical expression, look for opportunities to rationalize the denominator or simplify the radical itself. For instance, in the exercise \( \frac{6\sqrt{x}}{4x} \), we can divide both numerator and denominator by 2 to achieve the simpler form \( \frac{3\sqrt{x}}{2x} \). Simplicity in expressions prevents errors in further operations and provides a clearer view of the mathematical relationships at play.

Operations with Radicals

Performing operations with radicals can sometimes be challenging. Radicals in math are expressions that contain a root, such as a square root or cube root. There are several operations that you may need to perform with radicals, such as addition, subtraction, multiplication, division, and rationalization.

Consider these tips when working with radicals:

• Simplify radicals where possible by factoring out perfect squares or cubes.
• Combine radicals by adding or subtracting like terms — terms with the same radicand and index.
• Multiply radicals using the distributive property, especially when dealing with conjugates.
• Divide radicals by rationalizing the denominator, ensuring you do not leave a radical in the denominator.

In the exercise example, \( \frac{3}{2\sqrt{x}} \) involves division by a radical. The solution presented effectively demonstrates multiplication of the numerator and denominator by \(2\sqrt{x}\) to eliminate the radical in the denominator. Always remember to simplify the result further if possible, as with any operation involving radicals.