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Chapter 7 Radicals, Radical Functions, and Rational Exponents

Chapter 7 Radicals, Radical Functions, and Rational Exponents. § 7.1. Radical Expressions and Functions. Radicals. In this section, we introduce a new category of expressions and functions that contain roots . For example, the reverse operation of squaring a number is finding the

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Chapter 7 Radicals, Radical Functions, and Rational Exponents

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  1. Chapter 7Radicals, Radical Functions, and Rational Exponents

  2. §7.1 Radical Expressions and Functions

  3. Radicals In this section, we introduce a new category of expressions and functions that contain roots. For example, the reverse operation of squaring a number is finding the square root of the number. The symbol that we use to denote the principal square root is called a radical sign. The number under the radical sign is called the radicand. Together we refer to the radical sign and its radicand as a radical expression. Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.1

  4. Radical Expressions EXAMPLE Index of the Radical Radicand Radical Sign Radical Expression Blitzer, Intermediate Algebra, 5e – Slide #4 Section 7.1

  5. Radical Expressions P 487 Blitzer, Intermediate Algebra, 5e – Slide #5 Section 7.1

  6. Radical Expressions EXAMPLE Evaluate: SOLUTION The principal square root of a negative number, -16, is not a real number. Simplify the radicand. The principal square root of 169 is 13. Take the principal square root of 144, 12, and of 25, 5, and then add to get 17. Blitzer, Intermediate Algebra, 5e – Slide #6 Section 7.1

  7. Radical Expressions Check Point 1 on page 487 Evaluate: Principal Square Root means the answer is nonnegative Denotes the negative square root of a number Is a grouping symbol Blitzer, Intermediate Algebra, 5e – Slide #7 Section 7.1

  8. Radical Expressions Bottom of P 487 Blitzer, Intermediate Algebra, 5e – Slide #8 Section 7.1

  9. Radical Expressions See Figure 7.1 on page 488 P 488 Blitzer, Intermediate Algebra, 5e – Slide #9 Section 7.1

  10. Radical Functions EXAMPLE For the function, find the indicated function value: SOLUTION Substitute 4 for x in Simplify the radicand and take the square root of 9. Substitute 1 for x in Simplify the radicand and take the square root of 3. Blitzer, Intermediate Algebra, 5e – Slide #10 Section 7.1

  11. Radical Functions CONTINUED Substitute -1/2 for x in Simplify the radicand and take the square root. Substitute -1 for x in Simplify the radicand. The principal square root of a negative number is not a real number. Blitzer, Intermediate Algebra, 5e – Slide #11 Section 7.1

  12. Radical Functions Check Point 2 For the function, find the indicated function value: SOLUTION P 488 Blitzer, Intermediate Algebra, 5e – Slide #12 Section 7.1

  13. Radical Functions - Domain We have seen that the domain of a function f is the largest set of real numbers for which the value of f(x) is defined. Because only nonnegative numbers have real square roots, the domain of a square root function is the set of real numbers for which the radicand is nonnegative. In other words, we only use “allowable” x in the domain of the function. Not allowed for x is any value of x that would cause a negative number under a square root. Blitzer, Intermediate Algebra, 5e – Slide #13 Section 7.1

  14. Radical Functions - Domain EXAMPLE Find the domain of SOLUTION The domain is the set of real numbers, x, for which the radicand, 3x – 15, is nonnegative. We set the radicand greater than or equal to 0 and solve the resulting inequality. The domain of f is Blitzer, Intermediate Algebra, 5e – Slide #14 Section 7.1

  15. Radical Functions Check Point 3 Find the Domain of SOLUTION P 489 Blitzer, Intermediate Algebra, 5e – Slide #15 Section 7.1

  16. Radical Functions in Application EXAMPLE Police use the function to estimate the speed of a car, f(x), in miles per hour, based on the length, x, in feet, of its skid marks upon sudden braking on a dry asphalt road. Use the function to solve the following problem. A motorist is involved in an accident. A police officer measures the car’s skid marks to be 45 feet long. Estimate the speed at which the motorist was traveling before braking. If the posted speed limit is 35 miles per hour and the motorist tells the officer she was not speeding, should the officer believe her? Explain. Blitzer, Intermediate Algebra, 5e – Slide #16 Section 7.1

  17. Radical Functions in Application CONTINUED SOLUTION Use the given function. Substitute 45 for x. Simplify the radicand. Take the square root. The model indicates that the motorist was traveling at 30 miles per hour at the time of the sudden braking. Since the posted speed limit was 35 miles per hour, the officer should believe that she was not speeding. Blitzer, Intermediate Algebra, 5e – Slide #17 Section 7.1

  18. Radical Expressions The principal root is the positive root. P 490 Blitzer, Intermediate Algebra, 5e – Slide #18 Section 7.1

  19. Radical Expressions EXAMPLE Simplify each expression: SOLUTION The principal square root of an expression squared is the absolute value of that expression. In both exercises, it will first be necessary to express the radicand as an expression that is squared. (a) To simplify , first write as an expression that is squared: . Then simplify. Blitzer, Intermediate Algebra, 5e – Slide #19 Section 7.1

  20. Radical Expressions CONTINUED (b) To simplify , first write as an expression that is squared: . Then simplify. Blitzer, Intermediate Algebra, 5e – Slide #20 Section 7.1

  21. Radical Functions Check Point 5 Simplify each expression: P 490 Blitzer, Intermediate Algebra, 5e – Slide #21 Section 7.1

  22. Radical Expressions P 491 Blitzer, Intermediate Algebra, 5e – Slide #22 Section 7.1

  23. Radical Functions EXAMPLE For the function, find the indicated function value: SOLUTION Substitute 13 for x in Simplify the radicand and take the cube root of 27. Substitute 0 for x in Simplify the radicand and take the cube root of 1. Blitzer, Intermediate Algebra, 5e – Slide #23 Section 7.1

  24. Radical Functions CONTINUED Substitute -63 for x in Simplify the radicand and take the cube root of -125 and then simplify. Blitzer, Intermediate Algebra, 5e – Slide #24 Section 7.1

  25. Radical Expressions See Figure 7.2 on page 491 P 491 Blitzer, Intermediate Algebra, 5e – Slide #25 Section 7.1

  26. Radical Functions Check Point 6 For the function, find the indicated function value: P 492 Blitzer, Intermediate Algebra, 5e – Slide #26 Section 7.1

  27. Radical Expressions Blitzer, Intermediate Algebra, 5e – Slide #27 Section 7.1

  28. Radical Expressions EXAMPLE Simplify: SOLUTION Begin by expressing the radicand as an expression that is cubed: . Then simplify. We can check our answer by cubing -5x: By obtaining the original radicand, we know that our simplification is correct. Blitzer, Intermediate Algebra, 5e – Slide #28 Section 7.1

  29. Radical Functions Check Point 7 Simplify: P 492 Blitzer, Intermediate Algebra, 5e – Slide #29 Section 7.1

  30. Radical Expressions EXAMPLE Find the indicated root, or state that the expression is not a real number: SOLUTION because . An odd root of a negative real number is always negative. is not a real number because the index, 8, is even and the radicand, -1, is negative. No real number can be raised to the eighth power to give a negative result such as -1. Real numbers to even powers can only result in nonnegative numbers. Blitzer, Intermediate Algebra, 5e – Slide #30 Section 7.1

  31. Radical Expressions Blitzer, Intermediate Algebra, 5e – Slide #31 Section 7.1

  32. Radical Expressions EXAMPLE Simplify: SOLUTION Each expression involves the nth root of a radicand raised to the nth power. Thus, each radical expression can be simplified. Absolute value bars are necessary in part (a) because the index, n, is even. if n is even. if n is odd. Blitzer, Intermediate Algebra, 5e – Slide #32 Section 7.1

  33. Radical Functions Check Point 8 Find the indicated root, or state not real: P 493 Blitzer, Intermediate Algebra, 5e – Slide #33 Section 7.1

  34. Radical Functions Check Point 9 Simply: P 494 Blitzer, Intermediate Algebra, 5e – Slide #34 Section 7.1

  35. DONE

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