AATA Date 121013 SWBAT add and multiply radicals

AAT-A Date: 12/10/13 SWBAT add and multiply radicals Do Now: Rogawski #77 a get the page 224 Complete HW Requests: Adding Subtracting Multiplying Radicals Worksheets Continue Vocab sheet Closure-check answers Students will work pg 254 #43 -48 "Do not judge me by my successes, judge me by how many times I fell down and got back up again. “ Nelson Mandela HW: Complete Division of Radicals WS Announcements : Math Team Cancelled Wed. Tutoring: Tues. and Thurs. 3 -4 Martin-Gay, Developmental Mathematics 1

Simplifying Radical Expressions Martin-Gay, Developmental Mathematics 2

Rationalizing the Denominator Rationalizing the denominator -rewrite a radical quotient with the radical confined to ONLY the numerator. There is no radical in the denominator! Process: Multiply the quotient by a form of 1 to eliminate the radical in the denominator. Martin-Gay, Developmental Mathematics 3

Rationalizing the Denominator Example Rationalize the denominator. Martin-Gay, Developmental Mathematics 4

Conjugates To simplify rational quotients with a sum or difference of terms in a denominator, rather than a single radical. Process: Multiply by the conjugate of the numerator or denominator (which ever one we are rationalizing). The conjugate uses the same terms, but the opposite operation (+ or ). Martin-Gay, Developmental Mathematics 5

Rationalizing the Denominator Example Rationalize the denominator. Martin-Gay, Developmental Mathematics 6

§ 15. 4 Multiplying and Dividing Radicals

Multiplying and Dividing Radical Expressions If and are real numbers, Martin-Gay, Developmental Mathematics 8

Multiplying and Dividing Radical Expressions Example Simplify the following radical expressions. Martin-Gay, Developmental Mathematics 9

§ 15. 3 Adding and Subtracting Radicals

Sums and Differences Rules in the previous section allowed us to split radicals that had a radicand which was a product or a quotient. We can NOT split sums or differences. Martin-Gay, Developmental Mathematics 11

Like Radicals “like” terms- terms with the same variables raised to the same powers can be combined through addition and subtraction. Like radicals are radicals with the same index and the same radicand. Like radicals can be combined with addition or subtraction by using the distributive property. Martin-Gay, Developmental Mathematics 12

Adding and Subtracting Radical Expressions Example Can not simplify Martin-Gay, Developmental Mathematics 13

Adding and Subtracting Radical Expressions Example Simplify the following radical expression. Martin-Gay, Developmental Mathematics 14

Adding and Subtracting Radical Expressions Example Simplify the following radical expression. Martin-Gay, Developmental Mathematics 15

Adding and Subtracting Radical Expressions Example Simplify the following radical expression. Assume that variables represent positive real numbers. Martin-Gay, Developmental Mathematics 16

Square Roots Opposite of squaring a number is taking the square root of a number. A number b is a square root of a number a if b 2 = a. In order to find a square root of a, you need a # that, when squared, equals a. Martin-Gay, Developmental Mathematics 17

Principal Square Roots The principal (positive) square root is noted as The negative square root is noted as Martin-Gay, Developmental Mathematics 18

Radicands Radical expression is an expression containing a radical sign. Radicand is the expression under a radical sign. Note that if the radicand of a square root is a negative number, the radical is NOT a real number. Martin-Gay, Developmental Mathematics 19

th n Finding the root of a number t Finding the square root of a number involves finding a number that, when squared, equals the given number. t In other words, finding such that b 2 = a. t Some vocabulary involved with nth roots: n is the index of the expression. This is called a The index tells us what amount of radical symbol. factors we should look for in order to simplify a quantity. Examples: s is called the radicand of the radical If n = 3, we are looking for some expression. If the index n is even, then s value r such that r 3 = s. must be positive. This is because there is no If n = 4, we are looking for some value of r such that r 2 = -s. value r such that r 4 = s. Martin-Gay, Developmental Mathematics 20

nth Roots The nth root of a is defined as If the index, n, is even, the root is NOT a real number when a is negative. If the index is odd, the root will be a real number. * Martin-Gay, Developmental Mathematics 21

Radicands Example Martin-Gay, Developmental Mathematics 22

Perfect Squares Square roots of perfect square radicands simplify to rational numbers (numbers that can be written as a quotient of integers). Square roots of numbers that are not perfect squares (like 7, 10, etc. ) are irrational numbers. IF REQUESTED, you can find a decimal approximation for these irrational numbers. Otherwise, leave them in radical form. Martin-Gay, Developmental Mathematics 23

Perfect Square Roots Radicands might also contain variables and powers of variables. To avoid negative radicands, assume for this chapter that if a variable appears in the radicand, it represents positive numbers only. Example Martin-Gay, Developmental Mathematics 24

nth Roots Example Simplify the following. Martin-Gay, Developmental Mathematics 25

Cube Roots The cube root of a real number a Note: a is not restricted to non-negative numbers for cubes. Martin-Gay, Developmental Mathematics 26

Cube Roots Example Martin-Gay, Developmental Mathematics 27

Using the absolute value with radicals Let b = 1, then Now, let b = -1 but To make sure that the answer is positive we add an absolute value. If b is positive there is no problem, however, if b is negative we need |b| Martin-Gay, Developmental Mathematics 28

§ 15. 2 Simplifying Radicals

Product Rule and Quotient Rule for Square Roots If and are real numbers, Martin-Gay, Developmental Mathematics 30

Simplifying Radicals Simplify the following radical expressions. Factor radicand, isolate perfect squares, then simplify Example No perfect square factor, so the radical is already simplified. Martin-Gay, Developmental Mathematics 31

Simplifying Radicals Example Simplify the following radical expressions. Martin-Gay, Developmental Mathematics 32

Product and Quotient Rule for Radicals If and are real numbers, Martin-Gay, Developmental Mathematics 33

Simplifying Radicals Simplify the following radical expressions. Factor radicand, isolate perfect squares, then simplify Example Martin-Gay, Developmental Mathematics 34

§ 15. 5 Solving Equations Containing Radicals

Extraneous Solutions Power Rule (text only talks about squaring, but applies to other powers, as well). If both sides of an equation are raised to the same power, solutions of the new equation contain all the solutions of the original equation, but might also contain additional solutions. A proposed solution of the new equation that is NOT a solution of the original equation is an extraneous solution. Martin-Gay, Developmental Mathematics 36

Solving Radical Equations Example Solve the following radical equation. Substitute into the original equation. true So the solution is x = 24. Martin-Gay, Developmental Mathematics 37

Solving Radical Equations Example Solve the following radical equation. Substitute into the original equation. Does NOT check, since the left side of the equation is asking for the principal square root. So the solution is . Martin-Gay, Developmental Mathematics 38

Solving Radical Equations Steps for Solving Radical Equations 1) 2) 3) 4) Isolate one radical on one side of equal sign. Raise each side of the equation to a power equal to the index of the isolated radical, and simplify. (With square roots, the index is 2, so square both sides. ) If equation still contains a radical, repeat steps 1 and 2. If not, solve equation. Check proposed solutions in the original equation. Martin-Gay, Developmental Mathematics 39

Solving Radical Equations Example Solve the following radical equation. Substitute into the original equation. true So the solution is x = 2. Martin-Gay, Developmental Mathematics 40

Solving Radical Equations Example Solve the following radical equation. Martin-Gay, Developmental Mathematics 41

Solving Radical Equations Example continued Substitute the value for x into the original equation, to check the solution. true So the solution is x = 3. Martin-Gay, Developmental Mathematics false 42

Solving Radical Equations Example Solve the following radical equation. Martin-Gay, Developmental Mathematics 43

Solving Radical Equations Example continued Substitute the value for x into the original equation, to check the solution. false So the solution is . Martin-Gay, Developmental Mathematics 44

Solving Radical Equations Example Solve the following radical equation. Martin-Gay, Developmental Mathematics 45

Solving Radical Equations Example continued Substitute the value for x into the original equation, to check the solution. true So the solution is x = 4 or 20. Martin-Gay, Developmental Mathematics 46

§ 15. 6 Radical Equations and Problem Solving

The Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. (leg a)2 + (leg b)2 = (hypotenuse)2 Martin-Gay, Developmental Mathematics 48

Using the Pythagorean Theorem Example Find the length of the hypotenuse of a right triangle when the length of the two legs are 2 inches and 7 inches. c 2 = 22 + 72 = 4 + 49 = 53 c = inches Martin-Gay, Developmental Mathematics 49

The Distance Formula By using the Pythagorean Theorem, we can derive a formula for finding the distance between two points with coordinates (x 1, y 1) and (x 2, y 2). Martin-Gay, Developmental Mathematics 50

The Distance Formula Example Find the distance between ( 5, 8) and ( 2, 2). Martin-Gay, Developmental Mathematics 51

Chapter Sections 15. 1 – Introduction to Radicals 15. 2 – Simplifying Radicals 15. 3 – Adding and Subtracting Radicals 15. 4 – Multiplying and Dividing Radicals 15. 5 – Solving Equations Containing Radicals 15. 6 – Radical Equations and Problem Solving Martin-Gay, Developmental Mathematics 52