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Understanding Radical Expressions and Functions: Square Roots and Cube Roots Explained

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This section covers the fundamentals of radical expressions and functions, including radical notation and the properties of square roots and cube roots. It explains that every positive number has two square roots, and introduces key terms such as the radical sign and radicand. The section provides examples for evaluating square and cube roots, determining the domain of functions, and applying these concepts in real-world scenarios. This foundational knowledge will solidify your understanding of radical expressions in mathematics.

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Understanding Radical Expressions and Functions: Square Roots and Cube Roots Explained

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  1. Section 7.1 • Radical Expressions and Functions

  2. Objectives Radical Notation The Square Root Function The Cube Root Function

  3. Radical Notation • Every positive number a has two square roots, one positive and one negative. Recall that the positive square root is called the principal square root. • The symbol is called the radical sign. • The expression under the radical sign is called the radicand, and an expression containing a radical sign is called a radical expression. • Examples of radical expressions:

  4. Example Evaluate each square root. a. b. c.

  5. Example Approximate to the nearest thousandth. Solution

  6. Example Evaluate the cube root. a. b. c.

  7. Example Find each root, if possible. a. b. c. Solution a. b. c. An even root of a negative number is not a real number.

  8. Example Write each expression in terms of an absolute value. a. b. c. Solution a. b. c.

  9. Example If possible, evaluate f(1) and f(2) for each f(x). a. b. Solution a. b.

  10. Example Calculate the hang time for a ball that is kicked 75 feet into the air. Does the hang time double when a ball is kicked twice as high? Use the formula Solution The hang time is The hang time is The hang times is less than double.

  11. Example Find the domain of each function. Write your answer in interval notation. a. b. Solution Solve 3 – 4x 0. The domain is b. Regardless of the value of x; the expression is always positive. The function is defined for all real numbers, and it domain is

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