Problem 42 - College Algebra

Question

In Exercises \(39-48\), rationalize the denominator. $$\frac{\sqrt{7}}{\sqrt{3}}$$

Step-by-Step Solution

Answer

The rationalized form of \(\frac{\sqrt{7}}{\sqrt{3}}\) is \(\frac{\sqrt{21}}{3}\).

Step 1: Identify the denominator

In the given fraction, \(\frac{\sqrt{7}}{\sqrt{3}}\), \(\sqrt{3}\) is the denominator.

Step 2: Multiply by the denominator's radical

Multiply both the numerator and the denominator by the same radical, which here is \(\sqrt{3}\). This results into \(\frac{\sqrt{7}*\sqrt{3}}{\sqrt{3}*\sqrt{3}}\) which simplifies into \(\frac{\sqrt{21}}{3}\).

Step 3: Simplify the result

After multiplying, check if the fraction can be simplified further or not. Here, the result can't be simplified further.

Key Concepts

Radical Expressions

Radical expressions are elements in mathematics that involve square roots, cube roots, or other root forms. When dealing with these expressions, it’s essential to understand how they work and how they can be manipulated. In this context, consider \(\frac{\sqrt{7}}{\sqrt{3}}\).

• Here, \(\sqrt{7}\) and \(\sqrt{3}\) are both radical expressions.
• A radical expression can be either a part of a fraction (as the numerator or denominator) or can stand alone as an expression.

Radicals are most commonly represented using the square root symbol. To rationalize them, such as in our exercise, means to turn an expression involving a square root into an equivalent expression without radicals in the denominator.

Simplifying Fractions

Simplifying fractions is the process of reducing the fraction to its simplest form. This involves eliminating any common factors that exist between the numerator and the denominator.
In the given exercise, we worked on simplifying the fraction \(\frac{\sqrt{21}}{3}\).

• To simplify, look for any common factors between the numerator and the denominator.
• If common factors exist, divide them out from both parts of the fraction.

In the exercise, the simplified result \(\frac{\sqrt{21}}{3}\) cannot be reduced further since the numerator and the denominator have no further common factors apart from 1. Simple fractions and their simplification help in better understanding and usage in further calculations.

Algebraic Fractions

Algebraic fractions are fractions that have algebraic expressions, which can include variables and constants, in their numerators or denominators. Rationalizing a denominator usually involves transforming such a fraction into another equivalent fraction.

• For the fraction \(\frac{\sqrt{7}}{\sqrt{3}},\) we used multiplication by \(\sqrt{3}\) to eliminate the radical in the denominator.
• This is necessary because it's generally easier to work with fractions that do not have radicals in the denominator.

By rationalizing the denominator, we simplify calculations and align with conventional forms of representing algebraic fractions. This process is used widely in mathematics to make expressions easier and more standard to interpret.