Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations Square Roots вЂў A square root of a real number вЂњaвЂќ is a real number that multiplies by itself to give вЂњaвЂќ What is a square root of 9? 3 What is another square root of 9? пЂ 3 вЂў What is the square root of -4 ? Square root of вЂ“ 4 does not exist in the real number system вЂў Why is it that square roots of negative numbers do not exist in the real number system? No real number multiplied by itself can give a negative answer вЂў Every positive real number вЂњaвЂќ has two square roots that have equal absolute values, but opposite signs The two square roots of 16 are: The two square roots of 5 are: (Positive 16 and пЂ 16 5 and пЂ 5 Square Root : PRINCIPLE simplified ROOT) : 4 and пЂ 4 Even Roots (2,4,6,вЂ¦) вЂў The even вЂњnthвЂќ root of a real number вЂњaвЂќ is a real number that multiplies by itself вЂњnвЂќ times to give вЂњaвЂќ вЂў Even roots of negative numbers do not exist in the real number system, because no real number multiplied by itself an even number of times can give a negative number 4 пЂ 16 does not exist вЂў Every positive real number вЂњaвЂќ has two even roots that have equal absolute values, but opposite signs The fourth roots of 16: 4 16 and пЂ 4 16 simplified : 2 and пЂ 2 The fourth roots of 7: 4 7 and - 4 7 (Positive Even Root : PRINCIPLE ROOT) Radical Expressions вЂў On the previous slides we have used symbols of the form: n a вЂў This is called a radical expression and the parts of the expression are named: n Index: Radical Sign : Radicand: a вЂў Example: 5 8 Index : 5 Radicand : 8 Cube Roots вЂў The cube root of a real number вЂњaвЂќ is a real number that multiplies by itself 3 times to give вЂњaвЂќ вЂў Every real number вЂњaвЂќ has exactly one cube root that is positive when вЂњaвЂќ is positive, and negative when вЂњaвЂќ is negative Only cube root of вЂ“ 8: 3 пЂ 8 пЂЅ пЂ 2 3 6 Only cube root of 6: No such thing as a principle cube root! Odd Roots (3,5,7,вЂ¦) вЂў The odd nth root of a real number вЂњaвЂќ is a real number that multiplies by itself вЂњnвЂќ times to give вЂњaвЂќ вЂў Every real number вЂњaвЂќ has exactly one odd root that is positive when вЂњaвЂќ is positive, and negative when вЂњaвЂќ is negative The only fifth root of - 32: 3 пЂ 32 пЂЅ пЂ 2 The only fifth root of -7: пЂ5 7 Rational, Irrational, and Non-real Radical Expressions вЂў n a is non-real only if the radicand is negative and the index is even 6 вЂў n пЂ 20 is non - real because radicand and index is even only if the radicand a represents a rational number th can be written as a вЂњperfect n вЂќ power of an integer or the ratio of two integers 5 вЂў is negative n пЂ 32 is rational because пЂ 32 пЂЅ пЂЁпЂ 2 пЂ© 5 5 пЂ 32 пЂЅ пЂ 2 a represents an irrational number only if it is a real number and the radicand can not be written as вЂњperfect nthвЂќ power of an integer or the ratio of two integers . 4 8 is irrational of an integer because 8 is not the fourth power or the ratio of two integers 4 8 Homework Problems вЂў Section: 10.1 вЂў Page: 666 вЂў Problems: All: 1 вЂ“ 6, Odd: 7 вЂ“ 31, 39 вЂ“ 57, 65 вЂ“ 91 вЂў MyMathLab Homework Assignment 10.1 for practice вЂў MyMathLab Quiz 10.1 for grade Exponential Expressions an вЂњaвЂќ is called the base вЂњnвЂќ is called the exponent вЂў If вЂњnвЂќ is a natural number then вЂњanвЂќ means that вЂњaвЂќ is to be multiplied by itself вЂњnвЂќ times. Example: What is the value of 24 ? (2)(2)(2)(2) = 16 вЂў An exponent applies only to the base (what it touches) Example: What is the value of: - 34 ? - (3)(3)(3)(3) = - 81 Example: What is the value of: (- 3)4 ? (- 3)(- 3)(- 3)(- 3) = 81 вЂў Meanings of exponents that are not natural numbers will be discussed in this unit. Negative Exponents: a-n вЂў A negative exponent has the meaning: вЂњreciprocate the base and make the exponent positiveвЂќ n a пЂn пѓ¦1пѓ¶ пЂЅпѓ§ пѓ· пѓЁaпѓё Examples: 2 3 . пЂ2 пѓ¦2пѓ¶ пѓ§ пѓ· пѓЁ3пѓё 1 пѓ¦1пѓ¶ пЂЅ пѓ§ пѓ· пЂЅ 9 пѓЁ3пѓё пЂ3 3 27 пѓ¦3пѓ¶ пЂЅ пѓ§ пѓ· пЂЅ 8 пѓЁ2пѓё Quotient Rule for Exponential Expressions вЂў When exponential expressions with the same base are divided, the result is an exponential expression with the same base and an exponent equal to the numerator exponent minus the denominator exponent a m a n 5 4 5 7 пЂЅa mпЂn Examples: . x пЂЅ 5 4пЂ7 пЂЅ5 пЂ3 12 x 4 пЂЅ x 12 пЂ 4 пЂЅ x 8 Rational Exponents (a1/n) and Roots 1 вЂў An exponent of the form n has the meaning: вЂњthe nth root of the base, if it exists, and, if there are two nth roots, it means the principle (positive) oneвЂќ 1 a n , if it exists, is the n th root of a 1 (If there are two n th roots, a n is the principle (positive) 1 ( a n multiplies by itself n times to give a ) one) Examples of Rational Exponent of the Form: 1/n 1 100 2 пЂЅ 10 (positive square root of 100) 1 52 пЂЅ 1 пЂЁпЂ 3 пЂ© 2 5 (positive square root of 5) пЂЅ (Does not exist! ) 1 пЂ 32 пЂЅ пЂ 3 (negative square root of 3) 1 74 пЂЅ 1 пЂЁпЂ 9 пЂ© 7 1 пЂЁпЂ 8 пЂ© 6 пЂЅ пЂЅ 4 7 7 (positive пЂ 9 (seventh fourth root of 7) root of negative (Does not exist! ) 9) . Summary Comments about Meaning of a1/n вЂў When n is odd: вЂ“ a1/n always exists and is either positive, negative or zero depending on whether вЂњaвЂќ is positive, negative or zero вЂў When n is even: вЂ“ a1/n never exists when вЂњaвЂќ is negative вЂ“ a1/n always exists and is positive or zero depending on whether вЂњaвЂќ is positive or zero Rational Exponents of the Form: m/n вЂў An exponent of the form m/n has two equivalent meanings: (1) am/n means find the nth root of вЂњaвЂќ, then raise it to the power of вЂњmвЂќ (assuming that the nth root of вЂњaвЂќ exists) (2) am/n means raise вЂњaвЂќ to the power of вЂњmвЂќ then take the nth root of am (assuming that the nth root of вЂњamвЂќ exists) Example of Rational Exponent of the Form: m/n 82/3 by definition number 1 this means find the cube root of 8, then square it: 82/3 = 4 (cube root of 8 is 2, and 2 squared is 4) by definition number 2 this means raise 8 to the power of 2 and then cube root that answer: 82/3 = 4 (8 squared is 64, and the cube root of 64 is 4) Definitions and Rules for Exponents вЂў All the rules learned for natural number exponents continue to be true for both positive and negative rational exponents: 4 2 6 Product Rule: aman = am+n 37 пѓ—37 пЂЅ 37 2 Quotient Rule: am/an = am-n 37 4 пѓ—пЂЅ 3 пЂ 2 7 37 Negative Exponents: a-n = (1/a)n 4 пЂ 3 . 4 7 пѓ¦1пѓ¶ пЂЅ пѓ§ пѓ· пѓЁ3пѓё 7 Definitions and Rules for Exponents 2 Power Rules: пѓ¦ 4 пѓ§37 пѓ§ пѓЁ (am)n = amn (ab)m = ambm (a/b)m Zero Exponent: . = am / bm a0 = 1 (a not zero) 8 пѓ¶7 пѓ· пЂЅ 3 49 пѓ· пѓё 2 2 пЂЁ3 x пЂ© 7 2 пЂЅ 37 x 7 2 2 37 пѓ¦ 3 пѓ¶7 пѓ§ пѓ· пЂЅ 2 пѓЁ4пѓё 47 0 пѓ¦3пѓ¶ пѓ§ пѓ· пЂЅ 1 пѓЁ4пѓё вЂњSlide RuleвЂќ for Exponential Expressions вЂў When both the numerator and denominator of a fraction are factored then any factor may slide from the top to bottom, or vice versa, by changing the sign on the exponent Example: Use rule to slide all factors to other part of the fraction: m пЂn пЂr s a b c d r пЂЅ c d a пЂm пЂs b n вЂў This rule applies to all types of exponents вЂў Often used to make all exponents positive Simplifying Products and Quotients Having Factors with Rational Exponents вЂў All factors containing a common base can be combined using rules of exponents in such a way that all exponents are positive: вЂў Use rules of exponents to get rid of parentheses вЂў Simplify top and bottom separately by using product rules вЂў Use slide rule to move all factors containing a common base to the same part of the fraction вЂў If any exponents are negative make a final application of the slide rule Simplify the Expression: пѓ¦ пѓ¶ 3 8пѓ§ y y пѓ· пѓ§ пѓ· пѓЁ пѓё 1 3 2 пЂ1 6 y пЂ2 пЂ 2 3 1 7 6 пЂ y 9 пЂ1 y 12 2 пѓ—8 1 y4y 8y пЂ1 16 39 y 12 3 пЂ 2 2 3 пЂ1 7 1 пЂ 8 3 8y y y 8y 2 пЂ 4 пЂ 2 пЂ2 y 12 y 8 y 12 y 3 6 16 3 пЂ 2 12 7 32 y 12 y 12 Applying Rules of Exponents in Multiplying and Factoring пѓ¦ пЂ1 пѓ¶пѓ¦ 1 пѓ§ x 2 пЂ 2 пѓ·пѓ§ x 2 пЂ« x пѓ§ пѓ·пѓ§ пѓЁ пѓёпѓЁ вЂў Multiply: пЂЅ x пЂ 1 2 пЂ« 1 2 пЂ«x пЂ 1 пЂ 2 1 1 2 пЂ 2x 2 пЂ 2x пЂ пЂ 1 2 пЂ пЂ пЂ пЂ пѓ¶ 2 2 2 2 2 пѓ· пЂЅ x x пЂ« x x пЂ 2x пЂ 2x 2 пѓ· пѓё 1 1 1 1 1 2 1 пЂЅ x пЂ«x 0 пЂ1 пЂ 2x 2 пЂ 2x 1 пЂЅ 1пЂ« x пЂ1 пЂ 2x2 пЂ 2x вЂў Factor out the indicated factor: пЂЅ x пЂ 3 4 пЂЁ __ пЂ« __ пЂ© 4 пѓ¦ пѓ¶ 4пѓ§ 4 пѓ· пЂЅ x 5пЂ« x пѓ§ пѓ· пѓЁ пѓё пЂ 3 пЂЅ x пЂ 3 4 пЂ 5x пЂЁ5 пЂ« x пЂ© пЂ пЂ 1 1 1 2 1 2 3 1 4 пЂ« x4 ; x пЂ 3 4 Radical Notation вЂў Roots of real numbers may be indicated by means of either rational exponent notation or radical notation: n a is called a RADICAL is called a RADICAL (expressio n) SIGN n is called the INDEX a is called the RADICAND Notes About Radical Notation вЂў вЂў вЂў вЂў вЂў If no index is shown it is assumed to be 2 When index is 2, the radical is called a вЂњsquare rootвЂќ When index is 3, the radical is called a вЂњcube rootвЂќ When index is n, the radical is called an вЂњnth rootвЂќ In the real number system, we can only find even roots of non-negative radicands. There are always two roots when the index is even, but a radical with an even index always means the positive (principle) root вЂў We can always find an odd root of any real number and the result is positive or negative depending on whether the radicand is positive or negative Converting Between Radical and Rational Exponent Notation вЂў An exponential expression with exponent of the form вЂњm/nвЂќ can be converted to radical notation with index of вЂњnвЂќ, and vice versa, by either of the following formulas: m 1. a n 2 пЂЅ m пЂЅ n a 8 пЂЅ m пЂЁ aпЂ© n m пЂЅ 2 3 3 2 пЂЁ 8пЂ© 8 пЂЅ 8 3 2 3 64 пЂЅ 4 пЂЅ пЂЁ2 пЂ© пЂЅ 4 2 2. a вЂў These definitions assume that the nth root of вЂњaвЂќ exists n 3 Examples 4 5 пЂЅ 7 пЂЁ 5пЂ© 4 7 OR 7 5 4 9 5 8 пЂЅ 85 9 3 4x 11 пЂЅ 4 11 x 3 OR 4 пЂЁ xпЂ© 11 3 . n x n . вЂў If вЂњnвЂќ is even, then this notation means principle (positive) root: n x пЂЅ x n (absolute value needed to insure positive answer) вЂў If вЂњnвЂќ is odd, then: n x n x n пЂЅ x вЂў If we assume that вЂњxвЂќ is positive (which we often do) then we can say that: n пЂЅ x . Homework Problems вЂў Section: 10.2 вЂў Page: 675 вЂў Problems: All: 1 вЂ“ 10, Odd: 11 вЂ“ 47, 51 вЂ“ 97 вЂў MyMathLab Homework Assignment 10.2 for practice вЂў MyMathLab Quiz 10.2 for grade Product Rule for Radicals вЂў When two radicals are multiplied that have the same index they may be combined as a single radical having that index and radicand equal to the product of the two radicands: n a b пЂЅ n n ab 4 3 5 пЂЅ 3 пѓ— 5 пЂЅ 15 4 4 4 вЂў This rule works both directions: n ab пЂЅ n a n b 3 16 пЂЅ 3 8 2 пЂЅ 2 2 3 3 Quotient Rule for Radicals вЂў When two radicals are divided that have the same index they may be combined as a single radical having that index and radicand equal to the quotient of the two radicands 4 n 5 5 a a пЂЅ 4 пЂЅn 4 n 7 b 7 b вЂў This rule works both directions: n . a b n пЂЅ a 3 n b 5 8 3 пЂЅ 3 5 8 3 пЂЅ 5 2 Root of a Root Rule for Radicals вЂў When you take the mth root of the nth root of a radicand вЂњaвЂќ, it is the same as taking a single root of вЂњaвЂќ using an index of вЂњmnвЂќ . m n a пЂЅ 4 3 6 пЂЅ mn 12 a 6 NO Similar Rules for Sum and Difference of Radicals . n aпЂ« b п‚№ n 27 пЂ« 3 n n a пЂn b п‚№ 3 aпЂ«b 8 п‚№ 3 35 3пЂ« 2 п‚№ 3 35 n 3 aпЂb 27 пЂ 3 8 п‚№ 3 19 3пЂ2 п‚№ 3 19 Simplifying Radicals вЂў 1. 2. 3. 4. A radical must be simplified if any of the following conditions exist: Some factor of the radicand has an exponent that is bigger than or equal to the index There is a radical in a denominator (denominator needs to be вЂњrationalizedвЂќ) The radicand is a fraction All of the factors of the radicand have exponents that share a common factor with the index Simplifying when Radicand has Exponent Too Big 3 4 2 1. Use the product rule to write the single radical as a product of two radicals where the first radicand contains all factors whose exponents match the index and the second radicand contains all other factors 3 3 3 2 2 2. Simplify the first radical 3 2 2 Example Problem? 3 3 3 2 5 3 2 5 3 2 y 3 3 2y 3 24 x y 2 3x y Is there another exponent t hat is too big? Write this as a product of two radicals : 2 3x y 2 3x y 2 2 Simplify t he first radical : Simplifying when a Denominator Contains a Single Radical of Index вЂњnвЂќ 1. 2. Simplify the top and bottom separately to get rid of exponents under the radical that are too big Multiply the whole fraction by a special kind of вЂњ1вЂќ where 1 is in the form of: n m n m and m is the product of all the factors required make every exponent 3. in the radicand to be equal to "n" Simplify to eliminate the radical in the denominator Example 3 5 3 пЂЅ 4x y пЂЅ 3 6 5 5 пѓ— 3 пЂЅ 3 2 x y 3 2 2 y5 2 x y 5 3 6 3 5 2 2 x y 3 2 2 x y y 5 5 4 пЂЅ 4 3 пЂЅ 2 3 2 3 y5 2 x y 2 x y 3 2 4 5 5 5 35 2 x y y5 2 x y пЂЅ 5 пЂЅ 3 2 2 xy 2 35 2 x y 2 4 3 8x y 2 xy 2 4 Simplifying when Radicand is a Fraction 1. Use the quotient rule to write the single radical as a quotient of two radicals 2. Use the rules already learned for simplifying when there is a radical in a denominator Example 5 3 4 5 пЂЅ 5 3 4 5 пЂЅ 5 3 2 2 5 пЂЅ 5 3 2 2 5 пѓ— 5 2 3 2 3 5 пЂЅ 3пѓ—2 5 5 пЂЅ 2 3 5 24 2 Simplifying when All Exponents in Radicand Share a Common Factor with Index 1. Divide out the common factor from the index and all exponents 6 4 6 2 3 x y All exponents Dividing 3 2 8 3 in radicand all exponents 4 2 3 x y Problem? 2 пЂЅ 3 and index share what factor? in and index by 2 gives : 3 3 x 3 3 2 2 xy пЂЅ 3 x 3 4 xy 2 Simplifying Expressions Involving Products and/or Quotients of Radicals with the Same Index вЂў Use the product and quotient rules to combine everything under a single radical вЂў Simplify the single radical by procedures previously discussed Example 4 3 4 ab 4 ab 3 a b 4 пЂЅ 4 3 3 a b a 4 2 пЂЅ 4 a b 4 3 3 a b 4 пЂЅ пЂЅ a 3 a b a 4 b 4 пЂЅ 4 b a пЂЅ 4 b 4 a 3 4 a 4 a 3 Right Triangle вЂў A вЂњright triangleвЂќ is a triangle that has a 900 angle (where two sides intersect perpendicularly) c пЂЁhypotenuse пЂ© b 90 0 a вЂў The side opposite the right angle is called the вЂњhypotenuseвЂќ and is traditionally identified as side вЂњcвЂќ вЂў The other two sides are called вЂњlegsвЂќ and are traditionally labeled вЂњaвЂќ and вЂњbвЂќ Pythagorean Theorem вЂў In a right triangle, the square of the hypotenuse is always equal to the sum of the squares of the legs: c пЂЅ a пЂ«b 2 c b 90 0 a 2 2 Pythagorean Theorem Example вЂў It is a known fact that a triangle having shorter sides of lengths 3 and 4, and a longer side of length 5, is a right triangle with hypotenuse 5. 5 3 90 0 4 вЂў Note that Pythagorean Theorem is true: c пЂЅ a пЂ«b 2 2 2 5 пЂЅ 4 пЂ«3 25 пЂЅ 16 пЂ« 9 2 2 2 Using the Pythagorean Theorem вЂў We can use the Pythagorean Theorem to find the third side of a right triangle, when the other two sides are known, by finding, or estimating, the square root of a number Using the Pythagorean Theorem вЂў Given two sides of a right triangle with one side unknown: вЂ“ Plug two known values and one unknown value into Pythagorean Theorem вЂ“ Use addition or subtraction to isolate the вЂњvariable squaredвЂќ вЂ“ Square root both sides to find the desired answer Example вЂў Given a right triangle with a пЂЅ 7 and c пЂЅ 25 find the other side. c пЂЅ a пЂ«b 2 2 2 25 пЂЅ 7 пЂ« b 2 625 пЂЅ 49 пЂ« b 2 2 2 625 пЂ 49 пЂЅ 49 пЂ 49 пЂ« b 576 пЂЅ b 24 пЂЅ 576 пЂЅ b 2 2 Homework Problems вЂў Section: 10.3 вЂў Page: 685 вЂў Problems: Odd: 7 вЂ“ 19, 23 вЂ“ 57, 61 вЂ“ 107 вЂў MyMathLab Homework Assignment 10.3 for practice вЂў MyMathLab Quiz 10.3 for grade Adding and Subtracting Radicals вЂў Addition and subtraction of radicals can always be indicated, but can be simplified into a single radical only when the radicals are вЂњlike radicalsвЂќ вЂў вЂњLike RadicalsвЂќ are radicals that have exactly the same index and radicand, but may have different coefficients Which are like radicals? 3 4 4 3 5, 4 5, - 2 5 and 3 5 вЂў When вЂњlike radicalsвЂќ are added or subtracted, the result is a вЂњlike radicalвЂќ with coefficient equal to the sum or difference of the coefficients 3 4 5 пЂ«2 -2 4 5 пЂ« 3 5 пЂЅ 5 4 3 5 пЂЅ 4 5 Okay as is - can' t combine unlike radicals Note Concerning Adding and Subtracting Radicals вЂў When addition or subtraction of radicals is indicated you must first simplify all radicals because some radicals that do not appear to be like radicals become like radicals when simplified Example Not like terms (yet) 3 128 пЂ 5 пЂЅ 3 3 2 2 3 3 2пЂ«2 3 2 пЂ5 3 All like radicals пЂЅ 43 2 пЂ 5 3 2пЂ«4 пЂЅ 16 2пЂ«2 3 Simplify 3 3 3 2 пЂ5 7 3 3 2 пЂЅ3 3 2 individual 2пЂ«2 3 2 4 пЂЅ 2 пѓ— 23 2 пЂ 5 : 2 3 radicals : 2 3 2 пЂ« 2пѓ—2 3 2 Homework Problems вЂў Section: 10.4 вЂў Page: 691 вЂў Problems: Odd: 5 вЂ“ 57 вЂў MyMathLab Homework Assignment 10.4 for practice вЂў MyMathLab Quiz 10.4 for grade Simplifying when there is a Single Radical Term in a Denominator 1. Simplify the radical in the denominator 2. If the denominator still contains a radical, multiply the fraction by вЂњ1вЂќ where вЂњ1вЂќ is in the form of a вЂњspecial radicalвЂќ over itself 3. The вЂњspecial radicalвЂќ is one that contains the factors necessary to make the denominator radical factors have exponents equal to index 4. Simplify radical in denominator to eliminate it Example 3 2 3 Simplify denominato r: 9x 3 2 1 2 3 3 x 3 3 2 2 3 3 x 3 2 пѓ—3x 3 3 3 3 x 3 3x 3x Multiply by special "1" : 6x 3x 2 2 3 Use product rule : 2 Simplify denominato r: 2 Simplifying to Get Rid of a Binomial Denominator that Contains One or Two Square Root Radicals 1. Simplify the radical(s) in the denominator 2. If the denominator still contains a radical, multiply the fraction by вЂњ1вЂќ where вЂњ1вЂќ is in the form of a вЂњspecial binomial radicalвЂќ over itself 3. The вЂњspecial binomial radicalвЂќ is the conjugate of the denominator (same terms вЂ“ opposite sign) 4. Complete multiplication (the denominator will contain no radical) Example Radical in denominato 5 3пЂ« 3пЂ« Multiply fraction 3пЂ 2 Distribute 3пЂ 2 FOIL on bottom 2 5 пѓ— 2 r doesn' t need simplifyin by special one : on top : : 15 пЂ 10 9пЂ 4 Simplify bottom : 15 пЂ 10 3пЂ2 15 пЂ 10 g Homework Problems вЂў Section: 10.5 вЂў Page: 700 вЂў Problems: Odd: 7 вЂ“ 105 вЂў MyMathLab Homework Assignment 10.5 for practice вЂў MyMathLab Quiz 10.5 for grade Radical Equations вЂў An equation is called a radical equation if it contains a variable in a radicand вЂў Examples: 3 xпЂ xпЂ3 пЂЅ5 xпЂ« xпЂ«5 пЂЅ1 x пЂ« 4 пЂ 3 2x пЂЅ 0 Solving Radical Equations 1. Isolate ONE radical on one side of the equal sign 2. Raise both sides of equation to power necessary to eliminate the isolated radical 3. Solve the resulting equation to find вЂњapparent solutionsвЂќ 4. Apparent solutions will be actual solutions if both sides of equation were raised to an odd power, BUT if both sides of equation were raised to an even power, apparent solutions MUST be checked to see if they are actual solutions Why Check When Both Sides are Raised to an Even Power? вЂў Raising both sides of an equation to a power does not always result in equivalent equations вЂў If both sides of equation are raised to an odd power, then resulting equations are equivalent вЂў If both sides of equation are raised to an even power, then resulting equations are not equivalent (вЂњextraneous solutionsвЂќ may be introduced) вЂў Raising both sides to an even power, may make a false statement true: 2 2 4 4 пЂ 2 п‚№ 2 , however : пЂЁ- 2 пЂ© пЂЅ пЂЁ 2 пЂ© , пЂЁ- 2 пЂ© пЂЅ пЂЁ 2 пЂ© , etc. вЂў Raising both sides to an odd power never makes a false statement true: 3 3 5 5 пЂ 2 п‚№ 2 , and : пЂЁ- 2 пЂ© п‚№ пЂЁ 2 пЂ© , пЂЁ- 2 пЂ© п‚№ пЂЁ 2 пЂ© , etc. . Example of Solving Radical Equation xпЂ xпЂ3 пЂЅ5 xпЂ5пЂЅ пЂЁx пЂ 5пЂ© Check x пЂЅ 4 2 пЂЅ 4пЂ xпЂ3 пЂЁ xпЂ3 пЂ© 2 x пЂ 10 x пЂ« 25 пЂЅ x пЂ 3 2 x пЂ 11 x пЂ« 28 пЂЅ 0 2 пЂЁ x пЂ 4 пЂ©пЂЁ x пЂ 7 пЂ© пЂЅ 0 x пЂ 4 пЂЅ 0 OR xпЂ7пЂЅ0 x пЂЅ 4 OR x пЂЅ 7 4пЂ3 пЂЅ5? 4пЂ 1пЂЅ5? 3п‚№ 5 x пЂЅ 4 is NOT a solution Check x пЂЅ 7 7пЂ 7пЂ 7пЂ3 пЂЅ5? 4 пЂЅ5? 5пЂЅ5 xпЂЅ7 IS a solution Example of Solving Radical Equation xпЂ« пЂЁ x пЂ« 5 пЂЅ 1пЂ xпЂ«5 Check x пЂЅ 4 xпЂ«5 пЂЅ1 пЂ© 2 пЂЁ x пЂЅ 1пЂ x пЂ© 4пЂ« 2 x пЂ« 5 пЂЅ 1пЂ 2 x пЂ« x пЂЁпЂ 2 пЂ© 2 пЂЅ 9 пЂЅ1? 2пЂ«3 пЂЅ1? x пЂЅ 4 is NOT a solution x пЂЁ xпЂ© 4пЂЅ x 4пЂ« 5п‚№1 4 пЂЅ пЂ2 x пЂ2пЂЅ 4пЂ«5 пЂЅ1? 2 Equation has No Solution! пЃ¦ Example of Solving Radical Equation 3 x пЂ« 4 пЂ 3 2x пЂЅ 0 3 пЂЁ 3 xпЂ«4 пЂЅ xпЂ«4 3 пЂ© пЂЅпЂЁ 3 3 2x 2x пЂ© x пЂ« 4 пЂЅ 2x 4пЂЅ x (No need to check) 3 Homework Problems вЂў Section: 10.6 вЂў Page: 709 вЂў Problems: Odd: 7 вЂ“ 57 вЂў MyMathLab Homework Assignment 10.6 for practice вЂў MyMathLab Quiz 10.6 for grade

1/--страниц