The index of a mathematical expression refers to the exponent or power to which a base number or variable is raised. It represents the number of times a base is multiplied by itself and is a crucial concept in understanding operations involving square roots, higher roots, and rational exponents.
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The index of a square root is 2, as the base number is multiplied by itself once.
Higher roots, such as cube roots and fourth roots, have indices greater than 2, indicating the number of times the base is multiplied by itself.
Rational exponents, such as 3/2 or 5/4, have indices that are fractions, representing the relationship between the numerator and denominator.
The index of a radical expression determines the type of root being taken, with square roots having an index of 2, cube roots having an index of 3, and so on.
The index of a rational exponent is the denominator, while the numerator represents the power to which the base is raised.
Review Questions
Explain how the index of a square root relates to the operation of adding and subtracting square roots.
The index of a square root is 2, indicating that the base number is multiplied by itself once. When adding or subtracting square roots, the indices must be the same, as the roots are of the same degree. This allows for the combination of like terms and the application of the properties of square roots, such as the sum or difference of two perfect squares.
Describe the relationship between the index of a higher root and the process of evaluating higher roots.
The index of a higher root, such as a cube root or fourth root, determines the degree of the root being taken. A cube root has an index of 3, meaning the base number is multiplied by itself three times, while a fourth root has an index of 4. The index directly influences the steps required to evaluate and simplify higher root expressions, as the number of times the base is multiplied by itself must be taken into account.
Analyze how the index of a rational exponent affects the properties and operations involving rational exponents.
The index of a rational exponent, represented by the denominator, determines the root being taken of the base number. The numerator represents the power to which the base is raised. This relationship between the index and the numerator/denominator is crucial in understanding the properties of rational exponents, such as the product rule, quotient rule, and power rule. The index also dictates the steps required to simplify and evaluate expressions involving rational exponents.