Index

Definition

The index of a radical expression is the number that indicates the root being taken. It specifies the root, such as a square root, cube root, or fourth root, and is used to simplify and manipulate radical expressions.

5 Must Know Facts For Your Next Test

1. The index of a radical expression determines the type of root being taken, such as a square root (index 2), cube root (index 3), or fourth root (index 4).
2. Rational exponents can be used to represent radical expressions, where the exponent's numerator is the power and the denominator is the index of the root.
3. When simplifying radical expressions, the index is used to determine the appropriate operations, such as multiplying, dividing, or raising to a power.
4. The index is also crucial when adding, subtracting, and multiplying radical expressions, as the indices must be the same for the operations to be performed.
5. Solving radical equations often involves isolating the variable within a radical expression and then using the index to determine the appropriate steps to solve for the variable.

Review Questions

• Explain how the index of a radical expression is used to simplify expressions with roots.
  • The index of a radical expression determines the type of root being taken, such as a square root (index 2), cube root (index 3), or fourth root (index 4). When simplifying expressions with roots, the index is used to guide the appropriate operations. For example, to simplify $\sqrt[3]{27}$, the index of 3 indicates that a cube root is being taken, and the result would be 3. The index is crucial in determining how to manipulate radical expressions, whether it's multiplying, dividing, or raising to a power.
• Describe the relationship between the index of a radical expression and rational exponents.
  • Rational exponents can be used to represent radical expressions, where the exponent's numerator is the power and the denominator is the index of the root. For instance, $\sqrt[4]{x^2}$ can be written as $x^{2/4}$, where the index of 4 is represented by the denominator of the rational exponent. This relationship between the index and rational exponents allows for the simplification and manipulation of radical expressions using the laws of exponents, which is particularly useful when working with expressions containing both roots and powers.
• Analyze how the index of a radical expression is used to solve radical equations.
  • Solving radical equations often involves isolating the variable within a radical expression and then using the index to determine the appropriate steps to solve for the variable. For example, to solve the equation $\sqrt[3]{x + 5} = 7$, the index of 3 indicates that a cube root is being taken. By raising both sides of the equation to the power of 3 (the index), the radical can be eliminated, and the equation can be solved for the variable $x$. The index is crucial in this process, as it guides the necessary operations to isolate the variable and find the solution to the radical equation.

Related terms

Radical Expression: An expression containing a root, such as a square root, cube root, or any other root.

Radicand: The number or expression inside the radical sign, which is the value being rooted.

Rational Exponent: An exponent that can be expressed as a fraction, where the numerator is the power and the denominator is the index of the root.