Problem 45 - Algebra 1: Concepts and Skills

Question

Simplify the radical expression. $$ \frac{\sqrt{10}}{\sqrt{3}} $$

Step-by-Step Solution

Answer

The simplified radical expression is \(\sqrt{30}/3\).

Step 1: Use the rule \(\sqrt{a}/\sqrt{b} = \sqrt{a/b}\)

By using the rule, the given expression becomes \(\sqrt{10/3}\).

Step 2: Express the result in simplest radical form

Rationalize the denominator by multiplying both the numerator and denominator by the same number that will eliminate the square root in the denominator. Here, \(\sqrt{10/3} * \sqrt{3}/\sqrt{3} = \sqrt{30}/3\).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is a method used in algebra to eliminate radicals from the bottom of a fraction. The goal is to have a rational number in the denominator, which makes equations easier to handle and simplify.
To rationalize the denominator, multiply both the numerator and the denominator of the fraction by the same radical that appears in the denominator. This process removes the radical from the denominator because the product of a radical and itself is simply the number under the radical sign, making it rational.

• Always aim for a rational denominator.
• Multiply by the conjugate if dealing with complex radicals.

For example, to rationalize \( \frac{\sqrt{10}}{\sqrt{3}} \), multiply both the top and bottom by \( \sqrt{3} \). As a result, the expression becomes \( \frac{\sqrt{30}}{3} \). This new fraction has a rational denominator, achieving the goal of rationalization.

Simplifying Radicals

Simplifying radicals involves reducing the expression to its simplest form, where no square roots, cube roots, or any indicated root remains in the denominator.
Each part of the expression should be as simple as possible. To simplify, look for perfect square factors under the radical sign, since these can be taken out of the radical as whole numbers.

• Check for perfect square factors.
• Reduce any movable parts of the radical.

Using the example expression \( \sqrt{30} \), we can check for pairs of factors that form perfect squares. However, since 30 is composed of prime numbers (2, 3, 5), it cannot be simplified further. Thus, the simplest form is \( \sqrt{30} \). The objective is always to reach the simplest form that makes further calculations easier.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations such as addition, subtraction, multiplication, and division. These expressions represent mathematical relationships in a flexible form. They often include mathematical symbols and can include radical expressions like square roots.
Simplifying algebraic expressions requires an understanding of mathematical operations and rules.

• Use basic operations to simplify.
• Apply rules for exponents and radicals.

In the context of the exercise given, an algebraic expression involves manipulation and simplification of \( \frac{\sqrt{10}}{\sqrt{3}} \), demonstrating the rules of radicals and rationalization. Simplifying this expression required implementing the rules effectively to reach a more manageable and straightforward expression, \( \frac{\sqrt{30}}{3} \). Algebraic expressions are the foundation for building complex mathematical models.