1. C ollege A lgebra Basic Algebraic Operations (Appendix A) L:5 1 Instructor: Eng. Ahmed Abo absa University of Palestine IT-College

2. 2 Objectives Appendix A Cover the topics in Appendix A (A5 – A7): A-5 : Integer Exponents. A-7 : Radicals.

3. 3 A-5 : Integer Exponents Appendix A A-5 : contents 1 : Integer Exponents 2 : Scientific Notation

4. 1. Integer Exponents A-5 objectives.  Use the definition of exponents.  Simplify exponential expressions involving multiplying like bases, zero as an exponent, dividing like bases, negative exponents, raising a base to two exponents, raising a product to an exponent and raising a quotient to an exponent. 4

5. 1. Integer Exponents A-5 1.Forn a positive integer: a n =a ·a · … ·a n factors of a 2.Forn = 0, a 0 = 1a  0 0 0 is not defined 3.Forn a negative integer, a n = 1 a –n a  0 Definition of a n 3 =3. 3. 3. 3. 3 5 132 = 1 0 7 -5 = 1 7 -(-5) 1 7 5 = 5

6. A-5 1.a m a n =a m + n 2. () a nm =a mn 3.(ab) m =a m b m 4.       a b m = a m b m b  0 5. a m a n =a m–n = 1 a n–m a  0 Exponent Properties 1. Integer Exponents-Cont 1.a 5 a -7 =a -2 2. () a 3-2 =a -6 3.(ab) 3 =a 3 b 3 4.       a b 4 = a 4 b 4 b  0 5. a 3 a -2 =a 5 = 1 a -5 a  0 6

7. A-5 Example (1) *When mult. like bases you add your exponents Use the product rule to simplify the expression 7

8. A-5 Example (2) Evaluate. *Any expression raised to the 0 power simplifies to be 1 8

9. A-5 Example (3) Find the quotient *When div. like bases you subtract your exponents 9

10. A-5 Example (4) Simplify the exponential expression *When mult. like bases you add your exponents: 3 + (-5) = -2 *When div. like bases you subtract your exponents: -2 - (-20) = 18 10

11. A-5 2 : Scientific Notation objectives.  Write a number in scientific notation.  Write a number in standard notation, without exponents. 11

12. Scientific Notation A positive number is written in scientific notation if it is written in the form: where 1 < a < 10 and r is an integer power of 10. A-5 2 : Scientific Notation 12

13. A-5 2 : Scientific Notation Writing a Number in Scientific Notation Step 1: Move the decimal point so that you have a number that is between 1 and 10. In other words, you will put your decimal after the first non zero number Step 2: Count the number of decimal places moved in Step 1. If the decimal point was moved to the left, the count is positive. If the decimal point is moved to the right, the count is negative. Step 3: Write as a product of the number (found in Step 1) and 10 raised to the power of the count (found in Step 2). 13

14. A-5 2 : Scientific Notation Example (1) :Write the number in scientific notation: 483,000,000. Step 1: Move the decimal point so that you have a number that is between 1 and 10. *Decimal is at the end of the number *Move decimal to create a number between 1 and 10 Step 2: Count the number of decimal places moved in Step 1. How many decimal places did we end up moving? We started at the end of the number 483000000 and moved it between the 4 and 8. That looks like a move of 8 places. What direction did it move? Looks like we moved it to the left. So, our count is +8. Step 3: Write as a product of the number (found in Step 1) and 10 raised to the power of the count (found in Step 2). 14

15. A-5 2 : Scientific Notation Example (2) :Write the number in scientific notation: 0.00054. 15 Step 1: Move the decimal point so that you have a number that is between 1 and 10. *Decimal is at the beginning of the number *Move decimal to create a number between 1 and 10 Step 2: Count the number of decimal places moved in Step 1. How many decimal places did we end up moving? We started at the beginning of the number.00054 moved it between the 5 and 4. That looks like a move of 4 places. What direction did it move? Looks like we moved it to the right. So, our count is - 4. Step 3: Write as a product of the number (found in Step 1) and 10 raised to the power of the count (found in Step 2).

16. A-5 2 : Scientific Notation 16 Write a Scientific Number in Standard Form Basically, you just multiply the first number times the power of 10. Whenever you multiply by a power of 10, in essence what you are doing is moving your decimal place. If the power on 10 is positive, you move the decimal place that many units to the right. If the power on 10 is negative, you move the decimal place that many units to the left. Make sure you add in any zeros that are needed

17. A-5 2 : Scientific Notation 17 Example (1) :Write the number in standard notation, without exponents *Move the decimal 6 to the right Example (2) :Write the number in standard notation, without exponents *Move the decimal 5 to the left

18. 18 A-7 : Radicals Learning Objectives Appendix A After completing this section, you should be able to: 1.Find the principal nth root of an expression. 2.Find the nth root of an expression raised to the nth power. 3.Simplify radical expressions. 4.Multiply radicals that have the same index number. 5.Divide radicals that have the same index number. 6.Add and subtract like radicals. 7.Rationalize one term denominators of rational expressions. 8.Rationalize two term denominators of rational expressions.

19. 19 A-7 1.Principal nth root of an expression Definition of n th root radical If n is even: If n is even, then a and b must be nonnegative for the root to be a real number. If n is even and a is negative, then the root is not a real number. If n is odd: If n is odd, then a and b can be any real number.

20. 20 A-7 1.Principal nth root of an expression Things to note about radicals in general: When looking for the nth radical or nth root, you want the expression that, when you raise it to the nth power, you would get the radicand (what is inside the radical sign). When there is no index number, n, it is understood to be a 2 or square root. For example: = principal square root of x. Note that NOT EVERY RADICAL is a square root. If there is an index number n other than the number 2, then you have a root other than a square root.

21. 21 A-7 1.Principal nth root of an expression Example 1: Evaluate or indicate that the root is not a real number. The thought behind this is that we are looking for the square root of 100. This means that we are looking for a number that when we square it, we get 100. What do you think it is? Let’s find out if you are right: Since 10 squared is 100, 10 is the square root of 100. Note that we are only interested in the principal root and since 100 is positive and there is not a sign in front of the radical, our answer is positive 10. If there had been a negative in front of the radical our answer would have been -10.

22. 22 A-7 1.Principal nth root of an expression Example 2: Evaluate or indicate that the root is not a real number. Now we are looking for the negative of the fourth root of 16, which means we are looking for a number that when we raise it to the fourth power we get 16 (then we will take its negative). What do you think it is? Let’s find out if you are right: Since 2 raised to the fourth power is 16 and we are negating that, our answer is going to be -2. Note that the negative was on the outside of our even radical. If the negative had been on the inside of an even radical, then the answer would be no real number.

23. 23 A-7 1.Principal nth root of an expression Example 3: Evaluate or indicate that the root is not a real number. Now we are looking for the square root of -100, which means we are looking for a number that when we square it we get -100. What do you think it is? Let’s find out if you are right: Since there is no such real number that when we square it we get -100, the answer is not a real number.

24. 24 A-7 2. Find the nth root of an expression raised to the nth power. rule If n is an even positive integer, then If n is an odd positive integer, then If a problem does not indicate that a variable is positive, then you need to assume that we are dealing with both positive and negative real numbers and use this rule.

25. 25 A-7 2. Find the nth root of an expression raised to the nth power. Example 1: Simplify. Since it didn’t say that y is positive, we have to assume that it can be either positive or negative. And since the root number and exponent are equal, then we can use the rule. Since the root number and the exponent inside are equal and are the even number 2, we need to put an absolute value around y for our answer. The reason for the absolute value is that we do not know if y is positive or negative. So if we put y as our answer and it was negative, it would not be a true statement. For example if y was -5, then -5 squared would be 25 and the square root of 25 is 5, which is not the same as -5. The only time that you do not need the absolute value on a problem like this is if it stated that the variable is positive.

26. 26 A-7 2. Find the nth root of an expression raised to the nth power. Example 2: Simplify. Since it didn’t say that y is positive, we have to assume that it can be either positive or negative. And since the root number and exponent are equal, then we can use the rule. This time our root number and exponent were both the odd number 3. When an odd numbered root and exponent match then the answer is the base whether it is negative or positive.

27. 27 A-7 2. Simplifying a Radical Expression When you simplify a radical, you want to take out as much as possible. We can use the product rule of radicals (found below) in reverse to help us simplify the nth root of a number that we cannot take the nth root of as is, but has a factor that we can take the nth root of. If there is such a factor, we write the radicand as the product of that factor times the appropriate number and proceed. product rule of radicals We can also use the quotient rule of radicals (found below) to simplify a fraction that we have under the radical.quotient rule of radicals Note that the phrase "perfect square" means that you can take the square root of it. Just as "perfect cube" means we can take the cube root of the number, and so forth. I will be using that phrase in some of the following examples.

28. 28 A-7 2. Simplifying a Radical Expression A Product of Two Radicals With the Same Index Number In other words, when you are multiplying two radicals that have the same index number, you can write the product under the same radical with the common index number. Note that if you have different index numbers, you CANNOT multiply them together. Also, note that you can use this rule in either direction depending on what your problem is asking you to do.

29. 29 A-7 2. Simplifying a Radical Expression Example 1: Use the product rule to simplify. *Use the prod. rule of radicals to rewriteprod. rule of radicals Note that both radicals have an index number of 3, so we were able to put their product together under one radical keeping the 3 as its index number. Since we cannot take the cube root of 15 and 15 does not have any factors we can take the cube root of, this is as simplified as it gets.

30. 30 A-7 2. Simplifying a Radical Expression Example 2: Use the product rule to simplify. *Use the prod. rule of radicals to rewriteprod. rule of radicals Note that both radicals have an index number of 4, so we were able to put their product together under one radical keeping the 4 as its index number. Since we cannot take the fourth root of what is inside the radical sign and 24 does not have any factors we can take the fourth root of, this is as simplified as it gets.

31. 31 A-7 2. Simplifying a Radical Expression Example 3: Use the product rule to simplify. *Rewrite 75 as (25)(3) *Use the prod. rule of radicals to rewriteprod. rule of radicals *The square root of 25 is 5 Even though 75 is not a perfect square, it does have a factor that we can take the square root of. Check it out:

32. 32 A-7 2. Simplifying a Radical Expression Example 3: Use the product rule to simplify. *Rewrite as *Use the prod. rule of radicals to rewriteprod. rule of radicals *The square root of is Even though is not a perfect cube, it does have a factor that we can take the cube root of. Check it out:

33. 33 A-7 2. Simplifying a Radical Expression A Quotient of Two Radicals With the Same Index Number If n is even, x and y represent any nonnegative real number and y does not equal 0. If n is odd, x and y represent any real number and y does not equal 0. This works in the same fashion as the rule for a product of two radicals. This rule can also work in either direction.

34. 34 A-7 2. Simplifying a Radical Expression Example 1: Use the quotient rule to simplify *Use the quotient rule of radicals to rewrite quotient rule of radicals *The cube root of -1 is -1 and the cube root of 27 is 3

35. 35 A-7 2. Simplifying a Radical Expression Example 2: Use the quotient rule to simplify *Use the quotient rule of radicals to rewrite quotient rule of radicals *Simplify the fraction *Use the prod. rule of radicals to rewriteprod. rule of radicals *The square root of 4 x squared is 2|x| Since we cannot take the square root of 10 and 10 does not have any factors that we can take the square root of, this is as simplified as it gets

36. 36 A-7 2. Simplifying a Radical Expression Like Radicals Like radicals are radicals that have the same root number AND radicand (expression under the root). The following are two examples of two different pairs of like radicals

37. 37 A-7 2. Simplifying a Radical Expression Adding and Subtracting Radical Expressions Step 1: Simplify the radicals.Simplify the radicals. Step 2: Combine like radicals. You can only add or subtract radicals together if they are like radicals. like radicals You add or subtract them in the same fashion that you do like terms. Combine the numbers that are in front of the like radicals and write that number in front of the like radical part.

38. 38 A-7 2. Simplifying a Radical Expression Step 1: Simplify the radicals. Both radicals are as simplified as it gets. Step 2: Combine like radicals. Note how both radicals are the cube root of 2. These two radicals are like radicals. Example (1) : Add *Combine like radicals: 3x + 7x = 10x

39. 39 A-7 2. Simplifying a Radical Expression Step 1: Simplify the radicals. The 75 in the second radical has a factor that we can take the square root of. Can you think of what that factor is? Let’s see what we get when we simplify the second radical: Example (1) : Subtract *Rewrite 75 as (25)(3) *Use Prod. Rule of Radicals *Square root of 25 is 5 Step 2: Combine like radicals. *Combine like radicals: 4 - 30 = -26

40. 40 A-7 2. Simplifying a Radical Expression Step 1: Simplify the radicals. The 75 in the second radical has a factor that we can take the square root of. Can you think of what that factor is? Let’s see what we get when we simplify the second radical: Example (1) : Subtract *Rewrite 75 as (25)(3) *Use Prod. Rule of Radicals *Square root of 25 is 5 Step 2: Combine like radicals. *Combine like radicals: 4 - 30 = -26

41. 41 A-7 2. Simplifying a Radical Expression When a radical contains an expression that is not a perfect root, for example, the square root of 3 or cube root of 5, it is called an irrational number. So, in order to rationalize the denominator, we need to get rid of all radicals that are in the denominator Step 1: Multiply numerator and denominator by a radical that will get rid of the radical in the denominator. If the radical in the denominator is a square root, then you multiply by a square root that will give you a perfect square under the radical when multiplied by the denominator. If the radical in the denominator is a cube root, then you multiply by a cube root that will give you a perfect cube under the radical when multiplied by the denominator and so forth... Note that the phrase "perfect square" means that you can take the square root of it. Just as "perfect cube" means we can take the cube root of the number, and so forth. Rationalizing the Denominator (with one term)

42. 42 A-7 2. Simplifying a Radical Expression Keep in mind that as long as you multiply the numerator and denominator by the exact same thing, the fractions will be equivalent. Step 2: Simplify the radicals.Simplify the radicals. Step 3: Simplify the fraction if needed. Be careful. You cannot cancel out a factor that is on the outside of a radical with one that is on the inside of the radical. In order to cancel out common factors, they have to be both inside the same radical or be both outside the radical. Rationalizing the Denominator (with one term)

43. 43 A-7 2. Simplifying a Radical Expression Step 1: Multiply numerator and denominator by a radical that will get rid of the radical in the denominator. Step 2: Simplify the radicals. AND Step 3: Simplify the fraction if needed. Example (1) : Rationalize the denominator *Mult. num. and den. by sq. root of 5 *Den. now has a perfect square under sq. root

44. 44 A-7 2. Simplifying a Radical Expression Example (2) : Rationalize the denominator *Mult. num. and den. by cube root of *Den. now has a perfect cube under cube root *Cube root of 8 a cube is 2a

45. 45 A-7 2. Simplifying a Radical Expression Rationalizing the Denominator (with two terms) Step 1: Find the conjugate of the denominator. You find the conjugate of a binomial by changing the sign that is between the two terms, but keep the same order of the terms. a + b and a - b are conjugates of each other. Step 2: Multiply the numerator and the denominator of the fraction by the conjugate found in Step 1. Keep in mind that as long as you multiply the numerator and denominator by the exact same thing, the fractions will be equivalent. When you multiply conjugates together you get:

46. 46 A-7 2. Simplifying a Radical Expression Rationalizing the Denominator (with two terms) Step 3: Simplify the radicals. Step 4: Simplify the fraction if needed. Be careful. You cannot cancel out a factor that is on the outside of a radical with one that is on the inside of the radical. In order to cancel out common factors, they have to be both inside the same radical or be both outside the radical.

47. 47 A-7 2. Simplifying a Radical Expression Step 1: Find the conjugate of the denominator. In general the conjugate of a + b is a - b and vice versa. So what would the conjugate of our denominator be? It looks like the conjugate is Step 2: Multiply the numerator and the denominator of the fraction by the conjugate found in Step 1. Example (1) : Rationalize the denominator *Mult. num. and den. by conjugate of den. *Use distributive prop. to multiply the numerators *In general, product of conjugates is

48. 48 A-7 2. Simplifying a Radical Expression Step 3: Simplify the radicals. AND Step 4: Simplify the fraction if needed. *Square root of 3 squared is 3