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📘intermediate algebra review

key term - Common Index

Citation:

Definition

The common index refers to the shared exponent or power to which a set of radical expressions is raised. It is a crucial concept in the division of radical expressions, as it allows for the simplification and standardization of the radicals involved.

5 Must Know Facts For Your Next Test

  1. The common index is the highest exponent or power shared by all the radical expressions in the division problem.
  2. Identifying the common index is the first step in dividing radical expressions, as it allows for the standardization of the radicals.
  3. Once the common index is determined, the radical expressions can be divided by applying the rules of exponents.
  4. Dividing radical expressions with a common index often results in a simplified, rational expression.
  5. The common index is essential in maintaining the integrity of the radical expressions during the division process.

Review Questions

  • Explain the role of the common index in the division of radical expressions.
    • The common index plays a crucial role in the division of radical expressions. It represents the shared exponent or power to which the radicals are raised. By identifying the common index, the radical expressions can be standardized, allowing for the application of the rules of exponents to simplify the division process. The common index ensures that the integrity of the radical expressions is maintained throughout the division, leading to a simplified, rational expression as the result.
  • Describe the steps involved in dividing radical expressions with a common index.
    • When dividing radical expressions with a common index, the first step is to identify the highest exponent or power that is shared by all the radicals. This common index is then used to standardize the radicals by raising or lowering the individual exponents to match the common index. Once the radicals are standardized, the division can be performed by applying the rules of exponents. This process typically results in a simplified, rational expression that retains the essential characteristics of the original radical expressions.
  • Analyze the importance of the common index in the context of dividing radical expressions and simplifying the resulting expression.
    • The common index is a fundamental concept in the division of radical expressions, as it allows for the standardization and simplification of the radicals involved. By identifying the highest shared exponent or power, the common index enables the application of the rules of exponents, which is crucial for dividing the radical expressions and obtaining a simplified, rational result. Without the common index, the division process would be significantly more complex and prone to errors, as the radicals would not be in a standardized form. The common index is therefore an essential tool for maintaining the integrity of the radical expressions and ensuring the accuracy of the division and simplification steps.