Definition:

Rationalization is the process of simplifying and codifying mathematical expressions by removing fractional powers and simplifying radicals. It is a technique used in mathematics to simplify and manipulate expressions.

Steps involved in rationalization:

1. Find the common denominator: Identify the common denominator of the fractional parts of the expression.
2. Multiply by the common denominator: Multiply each fraction by the common denominator to obtain a common denominator.
3. Simplify the numerators: Simplify the numerators by finding common factors and reducing them to lowest terms.
4. Combine like terms: Combine like terms together.
5. Reduce fractions: If possible, reduce the resulting fractions to their lowest terms.

Examples:

1. Rationalize the following expression:

3/x + 5/x

Solution:

3/x + 5/x = (3 + 5) / x = 8/x

2. Rationalize the following expression:

√(4x)

Solution:

√(4x) = 2√x

Benefits of rationalization:

• Simplifies expressions by reducing fractional powers and simplifying radicals.
• Makes it easier to manipulate and simplify expressions.
• Can be used to find solutions to equations and inequalities.

Applications:

Rationalization is commonly used in:

• Algebra and calculus
• Trigonometry
• Statistics

Additional notes:

• Not all expressions can be rationalized.
• Some expressions may not be simplified further after rationalization.
• Rationalization is a technique that can be applied to a wide variety of mathematical expressions.