Note: In this section, the "undoing" or "reversing" of the squaring process will be introduced. What are the square roots of 16?

Transcription

1 Section 8.1 Video Guide Introduction to Square Roots Objectives: 1. Evaluate Square Roots 2. Determine Whether a Square Root is Rational, Irrational, or Not a Real Number 3. Find Square Roots of Variable Expressions Section 8.1 Objective 1: Evaluate Square Roots Part I Text Examples 1, 2, 3, and 4 Video Length 8:41 Note: In this section, the "undoing" or "reversing" of the squaring process will be introduced. Definition For any real numbers a and b, b is a of a if. For example, because 2 3 9, then 3 is a square root of 9 (Note: 3 is also a square root of 9). What are the square roots of 16? There are actually results is the "square root of 9." 3 9 The is the nonnegative square root. A Note: The side length of this square represents the principal square root of the area. Copyright 2018 Pearson Education, Inc. 267

2 Find the square roots of 49. What is the positive square root of 49? What are the square roots of 9 100? What is the positive square root of 9 100? Properties of Square Roots Every positive real number has, one positive and one negative. The square root of 0 is 0. In symbols,. We use the symbol, called a, to denote the square root of a real number. The nonnegative square root is called the square root. The number under the radical is called the. For example, the radicand in 25 is. 1. Example: Evaluate each square root: (a) 121 (a) 121 (b) (b) (c) 100 (c) Copyright 2018 Pearson Education, Inc.

3 Section 8.1 Objective 1: Evaluate Square Roots Part II Text Examples 5 and 6 Video Length 3:10 2. Example: Evaluate each expression: (a) 5 36 (a) 5 36 (b) 81 9 (b) 81 9 (c) (c) Caveat: Copyright 2018 Pearson Education, Inc. 269

4 Section 8.1 Objective 2: Determine Whether a Square Root is Rational, Irrational, or Not a Real Number Video Length 10:12 We are now going to look at some additional properties of square roots. More Properties of Square Roots The square root of a perfect square is a number. The square root of a positive rational number that is not a perfect square is an number. For example, 20 is an irrational number because 20 is not a perfect square. The square root of a negative real number is a number. For example, 2 is not a real number. 3. Example: Approximate 17 by writing it rounded to two decimal places. 4. Example: Determine if each square root is rational, irrational, or not a real number: (a) 11 (a) (b) 144 (b) (c) 54 (c) Note: Remember, the radicand is the number (or expression) under the radical. 270 Copyright 2018 Pearson Education, Inc.

5 Section 8.1 Objective 3: Find Square Roots of Variable Expressions Video Length 6:23 Find the following What about the following? Note: Be careful with this one. 2 x Definition For any real number a,. 5. Example: Simplify the following. (a) 81x 2 (a) 81x 2 (b) a 4 2 (b) a 4 2 Note: Observe the following restriction on the given variable and how it affects the final answer. (c) 2 x, x 0 (c) 2 x (d) p 2 2, p 2 (d) p 2 2 Copyright 2018 Pearson Education, Inc. 271

6 Section 8.2 Video Guide Simplifying Square Roots Objectives: 1. Use the Product Rule to Simplify Square Roots of Constants 2. Use the Product Rule to Simplify Square Roots of Variable Expressions 3. Use the Quotient Rule to Simplify Square Roots Section 8.2 Objective 1: Use the Product Rule to Simplify Square Roots of Constants Video Length 9:16 Definition A square root expression is if the radicand does not contain any factors that are perfect squares. For example, 8 is not simplified because We want to develop an approach for simplifying radicals that have factors that are perfect squares. Consider the following: This suggests that. Note: The work only suggests equality. However, it is not a proof! Product Rule of Square Roots If a and b are real numbers, then In other words, 272 Copyright 2018 Pearson Education, Inc.

7 1. Example: Simplify: 75 Write the steps in words Step 1 Show the steps with math Step 2 Step Example: Simplify: Copyright 2018 Pearson Education, Inc. 273

8 Section 8.2 Objective 2: Use the Product Rule to Simplify Square Roots of Variable Expressions Video Length 9:30 We know from the last section Assume a 0 : if. 2 a 4 a 6 a 8 a Note: To stay consistent with the restriction made above, we are going to assume that the variable(s) is/are positive for the following problems. 3. Example: Simplify: 10 49z 10 49z 4. Example: Simplify: 5 25b 5 25b 5. Example: Simplify: x y x y 274 Copyright 2018 Pearson Education, Inc.

9 Section 8.2 Objective 3: Use the Quotient Rule to Simplify Square Roots Video Length 6:33 Find the following: From the work above, we might conclude. Quotient Rule of Square Roots If a and b are nonnegative real numbers, b 0, then 6. Example: Simplify: Example: Simplify: Copyright 2018 Pearson Education, Inc. 275

10 8. Example: Simplify: 81z 4 w 8. Assume z is nonnegative and w is positive. 81z 4 w 8 9. Example: Simplify: 45m m, m m 2 5m 276 Copyright 2018 Pearson Education, Inc.

11 Section 8.3 Video Guide Adding and Subtracting Square Roots Objectives: 1. Add and Subtract Square Root Expressions with Like Square Roots 2. Add and Subtract Square Root Expressions with Unlike Square Roots Section 8.3 Objective 1: Add and Subtract Square Root Expressions with Like Square Roots Video Length 3:13 Definition Square root expressions are if each square root has the same. The idea behind adding like square roots is the same as the idea of combining like terms. 1. Example: Add or subtract, as indicated: (a) (b) 3 xyz 10 xyz 5 xyz xyz 10 xyz 5 xyz Note: The final answer can be simplified further. (c) Copyright 2018 Pearson Education, Inc. 277

12 Section 8.3 Objective 2: Add and Subtract Square Root Expressions with Unlike Square Roots Video Length 6:16 2. Example: Add: Example: Subtract: 3x 20x 7 5x 3, x 0 3x 20x 7 5x 3 Note: Remember, if x 0, then 2 x x. 2 x x. However, if we don't make any restrictions on x, then 278 Copyright 2018 Pearson Education, Inc.

13 Section 8.4 Video Guide Multiplying Expressions with Square Roots Objectives: 1. Find the Product of Square Roots Containing One Term 2. Find the Product of Square Roots Using the Distributive Property 3. Find the Product of Square Roots Using FOIL 4. Find the Product of Square Roots Using Special Products: A B 2, A B 2 A B A B Section 8.4 Objective 1: Find the Product of Square Roots Containing One Term Part I Text Examples 1, 2, and 3 Video Length 5:20 Earlier we learned how to use the Product Rule to simplify square roots. For example, 20, and In general a b Product Rule for Square Roots If a and b are nonnegative real numbers, then 1. Example: Multiply: (a) 5 7 (a) 5 7 (b) 6 2 (b) 6 2 (c) 2 5x 11 2x (c) 2 5x 11 2 x Copyright 2018 Pearson Education, Inc. 279

14 Section 8.4 Objective 1: Find the Product of Square Roots Containing One Term Part II Text Examples 4, 5, and 6 Video Length 9:48 2. Example: Multiply: 8z 6z 5 6 8z 6z 5 6 Squaring a Square Root 3. Example: Multiply: and for any real number a 0. (a) 2 13 (a) 2 13 (b) 34 2 (b) Example: Multiply: 4 12x 2 6x 4 12x 2 6 x 280 Copyright 2018 Pearson Education, Inc.

15 Section 8.4 Objective 2: Find the Product of Square Roots Using the Distributive Property Video Length 1:51 We are now going to use the Distributive Property with radicals. Recall, the Distributive Property states a b c 5. Example: Multiply: Copyright 2018 Pearson Education, Inc. 281

16 Section 8.4 Objective 3: Find the Product of Square Roots Using FOIL Video Length 4:10 Recall the FOIL method for multiplying binomials. For example, x 3 2x 5 6. Example: Multiply: (a) (b) 7 5w 2 5w 7 5w 2 5w 282 Copyright 2018 Pearson Education, Inc.

17 Section 8.4 Objective 4: Find the Product of Square Roots Using Special Products A B 2, A B 2, and A B A B Video Length 8:11 Recall the following special product formulas. For example, A B 2 A B 2 7. Example: Multiply: (a) (b) Copyright 2018 Pearson Education, Inc. 283

18 Do you remember the formula for the difference of two squares? Recall, A B A B 8. Example: Multiply: Note: He makes a really good point about these special product formulas and FOIL. If you forget the formulas, it is not the end of the world. You can use FOIL. 284 Copyright 2018 Pearson Education, Inc.

19 Section 8.5 Video Guide Dividing Expressions with Square Roots Objectives: 1. Find the Quotient of Two Square Roots 2. Rationalize a Denominator Containing One Term 3. Rationalize a Denominator Containing Two Terms Section 8.5 Objective 1: Find the Quotient of Two Square Roots Video Length 9:30 Thus far, we've looked at adding, subtracting, and multiplying square roots. Now we are going to focus on dividing expressions involving square roots. First we will come up with a definition for what it means for a square root to be simplified. Note: Remember, the radicand is the number (or expression) under the radical. Definition For a square root to be, the following requirements must be met: 1. The radicand any factors that are. 2. The radicand any. 3. No square root may appear as in a fraction. 1. Example: Simplify: Copyright 2018 Pearson Education, Inc. 285

20 2. Example: Copyright 2018 Pearson Education, Inc.

21 Section 8.5 Objective 2: Rationalize a Denominator Containing One Term Video Length 9:44 Recall that we do not allow radicals to occur in the denominator. So what we need to do is develop a method that allows us to rewrite an expression as an equivalent expression that does not contain a radical in the denominator. Consider Definition The process of rewriting a quotient in which the denominator contains a square root that is irrational as an equivalent quotient in which the denominator is rational is called. 3. Example: Rationalize the denominator: 1 7. Write the steps in words Step 1 Show the steps with math Step 2 Step Copyright 2018 Pearson Education, Inc. 287

22 4. Example: Rationalize the denominator: Example: Rationalize the denominator: 5pq 2r 4 4 5pq 2r Note: The final answer assumes a restriction on r. 288 Copyright 2018 Pearson Education, Inc.

23 Section 8.5 Objective 3: Rationalize a Denominator Containing Two Terms Part I Video Length 9:21 Now we are going to rationalize denominators containing two terms. In order to do this, we need to understand the following definition. Definition The of a binomial is a binomial having the same two terms with the sign of the second term changed. Examples: Why do we care so much about conjugates? Consider the following Note: He used FOIL to multiply the binomials above. However, you can also use the difference of 2 2 two squares formula: A B A B A B. 6. Example: Simplify: Write the steps in words Step 1 Show the steps with math Step 2 Step Copyright 2018 Pearson Education, Inc. 289

24 7. Example: Simplify: 2 n 3 2 n Copyright 2018 Pearson Education, Inc.

25 Section 8.5 Objective 3: Rationalize a Denominator Containing Two Terms Part II Video Length 4:10 8. Example: An answer which contains a square root is said to be 'the exact answer' when it contains a radical in simplified form. Irrational numbers do not have an exact decimal representation. Any decimal form of an answer containing irrational numbers is only an approximation. For the expression, complete parts (a) through (d). (a) Use your calculator to find the approximate value of the expression (Do not round until the final answer. Then round to four decimal places as needed.) (b) Rationalize the denominator to find the exact value of this expression (c) Use your calculator to find the approximate value of (b). (Do not round until the final answer. Then round to four decimal places as needed.) (d) Compare your results. o o The results are not the same. The results are the same. Copyright 2018 Pearson Education, Inc. 291

26 Section 8.6 Video Guide Solving Equations Containing Square Roots Objectives: 1. Determine Whether or Not a Number Is a Solution to a Radical Equation 2. Solve Equations Containing One Square Root 3. Solve Equations Containing Two Square Roots 4. Solve Problems Modeled by Radical Equations Section 8.6 Objective 1: Determine Whether or Not a Number Is a Solution to a Radical Equation Video Length 3:02 Note: Remember, the radicand is the expression under the radical. Definition When the variable in an equation occurs in a radicand, the equation is called a. Examples of radical equations: Definition A number is a to a radical equation if it satisfies the equation. 1. Example: Determine if x 2 is a solution of 6x Note: Write your final answer as a complete sentence. 292 Copyright 2018 Pearson Education, Inc.

27 Section 8.6 Objective 2: Solve Equations Containing One Square Root Part I Text Examples 2 and 3 Video Length 3:37 We are now going to solve radical equations containing a single square root. 2. Example: Solve: 2x Write the steps in words Step 1 Show the steps with math Step 2 Step 3 Step 4 Copyright 2018 Pearson Education, Inc. 293

28 Section 8.6 Objective 2: Solve Equations Containing One Square Root Part II Text Examples 4, 5, and 6 Video Length 11:10 Remember when we solved rational equations? There were times when we ended up with extraneous solutions. Extraneous solutions pop up with radical equations as well. 3. Example: Solve: x 12 x 4. Example: Solve: x 5 x Copyright 2018 Pearson Education, Inc.

29 Note: This is a good one! 5. Example: Solve: 7t Note: Pay attention to what he says about the yellow highlighted equation. Catching this early during the solving process can save a lot of time. Copyright 2018 Pearson Education, Inc. 295

30 Section 8.6 Objective 3: Solve Equations Containing Two Square Roots Video Length 10:33 We are now going to solve radical equations involving two square roots. Note: Take a deep breath...ready? 6. Example: Solve: 3y 1 y Copyright 2018 Pearson Education, Inc.

31 Section 8.6 Objective 4: Solve Problems Modeled by Radical Equations Video Length 3:11 7. Example: The annual rate of interest r (expressed as a decimal) required to have A dollars after 2 years from an initial deposit of P dollars is given by the equation A r 1. P Suppose you deposit $500 into an account that pays 2.9% annual interest. How much money will you have after two years? Note: Write your final answer as a complete sentence. Copyright 2018 Pearson Education, Inc. 297

32 Section 8.7 Video Guide Higher Roots and Rational Exponents Objectives: 1. Evaluate Higher Roots 2. Use Product and Quotient Rules to Simplify Higher Roots 3. Define and Evaluate Expressions of the Form 4. Define and Evaluate Expressions of the Form 5. Use Laws of Exponents to Simplify Expressions with Rational Exponents Section 8.7 Objective 1: Evaluate Higher Roots Part I Text Example 1 Video Length 7:35 Definition The of a number a, symbolized by, where n 2 is an integer, is defined as follows: 1 a n m n a means. In other words, the nth root of some number a means that For example, if you're asked to find 3 8, ask yourself Note: He doesn't actually write down the meaning of 4 16, but he says it. Make sure YOU write it down. If you're asked to find 4 16, what do you ask yourself? What number Note: Pay attention to what he says about the 'index'. 1. Example: Evaluate: (a) 121 (a) 121 (b) (b) Copyright 2018 Pearson Education, Inc.

33 The square root of a negative number The fourth root of a negative number For example, we know the fourth root of a negative number won't be because The fourth root of a negative number is not either because Consider n a. If n is, then. If n is, then For example, 4 12 is Copyright 2018 Pearson Education, Inc. 299

34 Section 8.7 Objective 1: Evaluate Higher Roots Part II Text Example 2 Video Length 9:46 Simplifying n a n If n 2 is a positive integer and a is a real number, then For example: if n 3 is if n 2 is. 2. Example: Simplify: (a) 4 4 x (a) 4 4 x (b) 3 27a 6 (b) a (c) y (c) y 300 Copyright 2018 Pearson Education, Inc.

35 Section 8.7 Objective 2: Use Product and Quotient Rules to Simplify Higher Roots Video Length 7:45 Note: In the beginning of the video, the Product Property of Radicals is mentioned. Here is the definition. The Product Property of Radicals If n a and n b are real numbers and n 2 is an integer, then n n n a b a b. We are now going to use this product property to help us simplify radical expressions. What do we mean when we say that a radical expression is simplified? Definition A radical expression is provided that the radicand does not contain any factors that are of the. 3. Example: Simplify the following: (a) 8 81c (a) 8 81c (b) x yz (b) xyz (c) x (c) x Note: He completed parts (a) (c), but skipped part (d). Go ahead and do it. You know you can. (d) (d) Copyright 2018 Pearson Education, Inc. 301

36 Quotient Property of Radicals If n a and n b are real numbers, b 0, and n 2 is an integer, then 4. Example: Simplify: n 3 4 n n 3 4 n 302 Copyright 2018 Pearson Education, Inc.

37 Section 8.7 Objective 3: Define and Evaluate Expressions of the Form Video Length 8:15 Up to this point, we have only worked with integer exponents. Now what we want to do is to define what it means for an expression to have a rational exponent. Consider 1/ a n We can conjecture that 1/n a. Definition of If a is a real number and n is an integer with n 2, then. Basically, the denominator of the rational exponent becomes. 5. Example: Write each of the following expressions as a radical and simplify, if possible. (a) (b) 1/2 16 (a) 1/4 w (b) 1/2 16 1/4 w (c) 3abc 1/5 (c) 3abc 1/5 6. Example: Rewrite each of the following radicals with a rational exponent. (a) 3 6x (a) 3 6x (b) c (b) c (c) 6 2x y 5 (c) 6 2x y 5 Note: Part (d) is already written with a rational exponent. However, it can be simplified further. (d) 64 2/3 (d) 64 2/3 Copyright 2018 Pearson Education, Inc. 303

38 Section 8.7 Objective 4: Define and Evaluate Expressions of the Form Video Length 8:40 mn / a Definition of If a is a real number, m/n is a rational number in with n 2, then m n a provided that n a exists. In other words, when you have a mn /, the denominator of the rational exponent. 7. Example: Evaluate each of the following expressions, if possible. (a) 2/3 16 (a) 2/3 16 (b) 3/4 w (b) 3/4 w (c) 3abc 2/5 (c) 3abc 2/5 Negative Exponent Rule If m n is a rational number, and if a is a nonzero number, then we define and if a Example: Rewrite each of the following with positive exponents and completely simplify, if possible. (a) (a) (b) (b) Copyright 2018 Pearson Education, Inc.

39 Section 8.7 Objective 5: Use Laws of Exponents to Simplify Expressions with Rational Exponents Video Length 12:46 Do you remember all the rules of integer exponents? All the laws of exponents that applied to integers also apply to rational numbers. The Law of Exponents If a and b are real numbers and if r and s are rational numbers, then assuming the expression is defined, Zero-Exponent Rule: Negative-Exponent Rule: Product Rule: Quotient Rule: Power Rule: Product to Power Rule: Quotient to Power Rule: Quotient to a Negative Power Rule: Definition The direction simplify shall mean the following: are. Each occurs. There are no in the expression. There are no written to. Copyright 2018 Pearson Education, Inc. 305

40 9. Example: Simplify the following: (a) 1/4 1/ /4 1/3 9 9 (b) y y 1/5 9/10 y y 1/5 9/ Example: Simplify the following: (a) y 4/5 1/3 4/5 y 1/3 (b) 4/9 1/9 4/9 2 6a 3a 4/9 1/9 4/9 2 6a 3a (c) 25x 2/5 y 1 1/2 1/2 2/5 1 25x y 306 Copyright 2018 Pearson Education, Inc.