Section 9.1- Radical Expressions and Graphs

Transcription

1 Chapter 9 Section 9.1- Radical Expressions and Graphs Objective: 1. Find square roots. 2. Decide whether a given root is rational, irrational, or not a real number. 3. Find cube, fourth, and other roots. 4. Use a calculator to find roots. Find square roots. Find all square roots of 49. To find the square roots of 49, think of a number than when multiplied by itself gives 49. On square root is 7 because 7 7 = 49. Another square root of 49 is 7 because ( 7)( 7) = 49. The number 49 has two square roots, 7 and 7; one is positive and one is negative. The positive or principal square root of a number is written with the symbol For example, the positive square root of 121 is 11, written The - is used for the negative square root of a number. For example, the negative square root of 121 is 11, written The symbol is called a radical sign, always represents the positive square root. 1 P a g e

2 The number inside the radical sign is called the radicand, and the entire expression, radical sign and radicand, is called a radical. An algebraic expression containing a radical is called a radical expression. We know that so the reverse of this process the since since since P a g e

3 Ex. Find each square root Ex. Find the square of each radical expression - 3 P a g e

4 Decide whether a given root is rational, irrational, or not a real number. All numbers with square roots that are rational are called perfect squares. Ex. Tell whether each square root is rational, irrational, or not a real number. 4 P a g e

5 Find cube, fourth, and other roots Finding the square root of a number is the inverse (reverse) of squaring a number. There are inverses to finding the cube of a number, or finding the fourth or higher power of a number. These inverses are the cube root and fourth root. Ex. Find each cube root. 5 P a g e

6 When a radical has an even index (square root, fourth root, and so on), the radicand must be nonnegative to yield a real number root. Ex. Find each root. 6 P a g e

7 Use a calculator to find roots. Radical expressions often represent irrational numbers. To find approximations of such radicals, we usually use a calculator. For example, where the symbol means is approximately equal to. Ex. Use a calculator to verify that each approximation is correct. Section 9.2 Rational Expressions 7 P a g e

8 Objectives 1. Use exponential notation for nth roots. 2. Define and use expressions of the form a m/n. 3. Convert between radicals and rational exponents. 4. Use the rules for exponents with rational exponents. Rational Exponents x x x x x x x x n x x 1 n Recall rules n can be any index when a is nonnegative n must be odd when a is negative 8 P a g e

9 Ex: = = = - = What about? 9 P a g e

10 CAUTION Notice the difference between parts (c) and (d) in Example 1. The radical in part (c) is the negative fourth root of a positive number, while the radical in part (d) is the principal fourth root of a negative number, which is not a real number. Definition of a m/n 10 P a g e

11 Definition: If a is a real number, m is an integer, and n is a positive integer greater than 1, then a m n n m a n a m where m is the power you are raising and n is the root you are taking. (Note: We assume that a is nonnegative if n is even.) Recall the Properties of Exponents that also hold true for rational numbers as exponents: For nonzero real numbers a and b and rational numbers m and n, 1 a a ( a is any real number.) a 0 1 m n 3. a a 4. a a m n a a m n m n m m n 5. a a n n 1 6. a n a and n n n 7. a b a b n a a 8. n b b n 1 a n a n In order for an exponential expression involving products and/or quotients to be SIMPLIFIED Each base must occur only once. 2. Each base must be raised to a single power. 3. All the powers must be positive. 11 P a g e

12 Ex: = ( ) = = = 12 P a g e

13 ( ) = ( ) = ( )= 13 P a g e

14 Alternative Definition of a m/n Converting between Rational Exponents and Radicals Ex: 15 1/2 4n 2/3 14 P a g e

15 7h 3/4 (2h) 2/5 g 4/5 10 5/6 15 P a g e

16 Applying Rules for Rational Exponents Write with only positive exponents. Assume that all variables represent positive real numbers. Do not make the common mistake of multiplying exponents in the first step. 6 3/4 6 1/2 16 P a g e

17 Applying Rules for Rational Exponents Rewrite all radicals as exponentials, and then apply the rules for rational exponents. Leave answers in exponential form. Assume that all variables represent positive real numbers. 17 P a g e

18 Section 9.3 Simplifying Rational Expressions Objectives: 1. Use the product rule for radicals. 2. Use the quotient rule for radicals. 3. Simplify radicals. 4. Use the Pythagorean formula. 5. Use the distance formula. Use the Product Rule for Radicals Ex: 18 P a g e

19 19 P a g e

20 Use the Quotient Rule for Radicals 20 P a g e

21 Simplifying Radicals 21 P a g e

22 Simplifying Radicals with Variables 22 P a g e

23 Pythagorean Formula The Pythagorean formula relates lengths of the sides of a right triangle Ex: 23 P a g e

24 The Distance Formula The distance formula, which allows us to compute the distance between two points in the coordinate plane is derived from the Pythagorean formula. Ex: Find the distance between (1, 6) and (4, 2). 24 P a g e

25 Section 9.7- Complex Numbers Objective: 1. Simplifying numbers of the form, where b >0. Introduction to Complex Numbers Radicals such as 4 is not a real number because no real number squared produces a negative number. In general, n x for any even n and x is not a real number. However, we can simplify using complex numbers. 2 i 1 and i P a g e

26 Examples Ex P a g e

27 The standard form of a complex number is a + bi. Where a is the real part and b is the imaginary part. Identify the real and imaginary part of each complex number. 4 2i -3i 5 27 P a g e

28 Adding and Subtracting Complex Numbers To add/subtract complex numbers you combine the real parts together and then combine the imaginary parts together. Ex. (-5 7i) + (-2 2i)= Ex. 2 4i 4 6i 8 5i = 28 P a g e

29 Ex = Ex = 29 P a g e

30 Multiplying Square Roots of Negative Numbers Ex. = Ex. = 30 P a g e

31 Dividing Square Roots of Negative Numbers Ex. = Ex. = 31 P a g e