1. R.7 Radical Notation and Rational Exponents  Simplify radical expressions.  Rationalize denominators or numerators in rational expressions.  Convert between exponential and radical notation.  Simplify expressions with rational exponents. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

2. Slide R.7 - 2 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Notation A number c is said to be a square root of a if c 2 = a. Similarly, c is a cube root of a if c 3 = a. nth Root A number c is said to be an nth root of a if c n = a. The symbol denotes the nth root of a. The symbol is called a radical. The number n is called the index. Any positive number has two square roots, one positive and one negative. For any even index, a positive number has two real-number roots. The positive root is called the principal root.

3. Slide R.7 - 3 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Examples Simplify each of the following: a)= 7, because 7 2 = 49. b) because c) because (  4) 3 =  64. d) is not a real number.

4. Slide R.7 - 4 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Properties of Radicals Let a and b be any real numbers or expressions for which the given roots exist. For any natural numbers m and n (n  1): 1. If n is even, 2. If n is odd, 3. 4. 5.

5. Slide R.7 - 5 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Examples a) b) c) d) e) f) g) h)

6. Slide R.7 - 6 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Another Example Perform the operation.

7. Slide R.7 - 7 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley The Pythagorean Theorem The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse: a 2 + b 2 = c 2. b c a

8. Slide R.7 - 8 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Juanita paddled her canoe across a river 525 feet wide. A strong current carried her canoe 810 feet downstream as she paddled. Find the distance Juanita actually paddled, to the nearest foot. Solution: 525 ft 810 ft x

9. Slide R.7 - 9 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Rationalizing Denominators or Numerators Rationalizing the denominator (or numerator) is done by multiplying by 1 in such a way as to obtain a perfect nth power. Example Rationalize the denominator.

10. Slide R.7 - 10 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Rationalizing Denominators or Numerators Conjugates Pairs of expressions of the form are called conjugates. The product of a pair contains no radicals and can be used to rationalize a denominator or numerator.

11. Slide R.7 - 11 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Rationalizing Denominators Example Rationalize the deominator.

12. Slide R.7 - 12 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Rational Exponents For any real number a and any natural numbers m and n, for which exists,

13. Slide R.7 - 13 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Examples Convert to radical notation and, if possible, simplify. a) b) c)

14. Slide R.7 - 14 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley More Examples Convert each to exponential notation. a) b) Covert to radical form and simplify. a) Simplify b)