2 Square RootsA square root of a real number “a” is a real number that multiplies by itself to give “a”What is a square root of 9?What is another square root of 9?What is the square root of -4 ?Square root of – 4 does not exist in the real number systemWhy 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 answerEvery positive real number “a” has two square roots that have equal absolute values, but opposite signsThe two square roots of 16 are:The two square roots of 5 are:
3 Even Roots (2,4,6,…)The even “nth” root of a real number “a” is a real number that multiplies by itself “n” times to give “a”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 numberEvery positive real number “a” has two even roots that have equal absolute values, but opposite signsThe fourth roots of 16:The fourth roots of 7:
4 Radical ExpressionsOn the previous slides we have used symbols of the form:This is called a radical expression and the parts of the expression are named:Index:Radical Sign :Radicand:Example:
5 Cube RootsThe cube root of a real number “a” is a real number that multiplies by itself 3 times to give “a”Every real number “a” has exactly one cube root that is positive when “a” is positive, and negative when “a” is negativeOnly cube root of – 8:Only cube root of 6:
6 Odd Roots (3,5,7,…)The odd nth root of a real number “a” is a real number that multiplies by itself “n” times to give “a”Every real number “a” has exactly one odd root that is positive when “a” is positive, and negative when “a” is negativeThe only fifth root of - 32:The only fifth root of -7:
7 Rational, Irrational, and Non-real Radical Expressions is non-real only if the radicand is negative and the index is evenrepresents a rational number only if the radicand can be written as a “perfect nth” power of an integer or the ratio of two integersrepresents an irrational number only if it is a real number and the radicand can not be written as “perfect nth” power of an integer or the ratio of two integers.
9 Exponential Expressions an“a” is called the base“n” is called the exponentIf “n” is a natural number then “an” means that “a” is to be multiplied by itself “n” times.Example: What is the value of 24 ?(2)(2)(2)(2) = 16An exponent applies only to the base (what it touches)Example: What is the value of: ?- (3)(3)(3)(3) = - 81Example: What is the value of: (- 3)4 ?(- 3)(- 3)(- 3)(- 3) = 81Meanings of exponents that are not natural numbers will be discussed in this unit.
10 Negative Exponents: a-n A negative exponent has the meaning: “reciprocate the base and make the exponent positive”Examples:.
11 Quotient Rule for Exponential Expressions 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 exponentExamples:.
12 Rational Exponents (a1/n) and Roots An exponent of the formhas the meaning: “the nth root of the base, if it exists, and, if there are two nth roots, it means the principle (positive) one”
13 Examples of Rational Exponent of the Form: 1/n .
14 Summary Comments about Meaning of a1/n When n is odd:a1/n always exists and is either positive, negative or zero depending on whether “a” is positive, negative or zeroWhen n is even:a1/n never exists when “a” is negativea1/n always exists and is positive or zero depending on whether “a” is positive or zero
15 Rational Exponents of the Form: m/n An exponent of the form m/n has two equivalent meanings:(1) am/n means find the nth root of “a”, then raise it to the power of “m”(assuming that the nth root of “a” exists)(2) am/n means raise “a” to the power of “m” then take the nth root of am(assuming that the nth root of “am” exists)
16 Example of Rational Exponent of the Form: m/n 82/3by 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:(8 squared is 64, and the cube root of 64 is 4)
17 Definitions and Rules for Exponents All the rules learned for natural number exponents continue to be true for both positive and negative rational exponents:Product Rule: aman = am+nQuotient Rule: am/an = am-nNegative Exponents: a-n = (1/a)n.
18 Definitions and Rules for Exponents Power Rules: (am)n = amn(ab)m = ambm(a/b)m = am / bmZero Exponent: a0 = 1 (a not zero).
19 “Slide Rule” for Exponential Expressions 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 exponentExample: Use rule to slide all factors to other part of the fraction:This rule applies to all types of exponentsOften used to make all exponents positive
20 Simplifying Products and Quotients Having Factors with Rational Exponents All factors containing a common base can be combined using rules of exponents in such a way that all exponents are positive:Use rules of exponents to get rid of parenthesesSimplify top and bottom separately by using product rulesUse slide rule to move all factors containing a common base to the same part of the fractionIf any exponents are negative make a final application of the slide rule
22 Applying Rules of Exponents in Multiplying and Factoring Factor out the indicated factor:
23 Radical NotationRoots of real numbers may be indicated by means of either rational exponent notation or radical notation:
24 Notes About Radical Notation If no index is shown it is assumed to be 2When index is 2, the radical is called a “square root”When index is 3, the radical is called a “cube root”When index is n, the radical is called an “nth root”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) rootWe 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
25 Converting Between Radical and Rational Exponent Notation An exponential expression with exponent of the form “m/n” can be converted to radical notation with index of “n”, and vice versa, by either of the following formulas:1.2.These definitions assume that the nth root of “a” exists
29 Product Rule for Radicals 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:This rule works both directions:
30 Quotient Rule for Radicals 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 radicandsThis rule works both directions:.
31 Root of a Root Rule for Radicals When you take the mth root of the nth root of a radicand “a”, it is the same as taking a single root of “a” using an index of “mn”.
32 NO Similar Rules for Sum and Difference of Radicals .
33 Simplifying RadicalsA 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 indexThere is a radical in a denominator (denominator needs to be “rationalized”)The radicand is a fractionAll of the factors of the radicand have exponents that share a common factor with the index
34 Simplifying when Radicand has Exponent Too Big 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 factorsSimplify the first radical
36 Simplifying when a Denominator Contains a Single Radical of Index “n” Simplify the top and bottom separately to get rid of exponents under the radical that are too bigMultiply the whole fraction by a special kind of “1” where 1 is in the form of:Simplify to eliminate the radical in the denominator
38 Simplifying when Radicand is a Fraction Use the quotient rule to write the single radical as a quotient of two radicalsUse the rules already learned for simplifying when there is a radical in a denominator
40 Simplifying when All Exponents in Radicand Share a Common Factor with Index Divide out the common factor from the index and all exponents
41 Simplifying Expressions Involving Products and/or Quotients of Radicals with the Same Index Use the product and quotient rules to combine everything under a single radicalSimplify the single radical by procedures previously discussed
43 Right TriangleA “right triangle” is a triangle that has a 900 angle (where two sides intersect perpendicularly)The side opposite the right angle is called the “hypotenuse” and is traditionally identified as side “c”The other two sides are called “legs” and are traditionally labeled “a” and “b”
44 Pythagorean TheoremIn a right triangle, the square of the hypotenuse is always equal to the sum of the squares of the legs:
45 Pythagorean Theorem Example 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.Note that Pythagorean Theorem is true:
46 Using the Pythagorean Theorem 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
47 Using the Pythagorean Theorem Given two sides of a right triangle with one side unknown:Plug two known values and one unknown value into Pythagorean TheoremUse addition or subtraction to isolate the “variable squared”Square root both sides to find the desired answer
48 ExampleGiven a right triangle with find the other side.
50 Adding and Subtracting Radicals Addition and subtraction of radicals can always be indicated, but can be simplified into a single radical only when the radicals are “like radicals”“Like Radicals” are radicals that have exactly the same index and radicand, but may have different coefficientsWhich are like radicals?When “like radicals” are added or subtracted, the result is a “like radical” with coefficient equal to the sum or difference of the coefficients
51 Note Concerning Adding and Subtracting Radicals 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
54 Simplifying when there is a Single Radical Term in a Denominator Simplify the radical in the denominatorIf the denominator still contains a radical, multiply the fraction by “1” where “1” is in the form of a “special radical” over itselfThe “special radical” is one that contains the factors necessary to make the denominator radical factors have exponents equal to indexSimplify radical in denominator to eliminate it
56 Simplifying to Get Rid of a Binomial Denominator that Contains One or Two Square Root Radicals Simplify the radical(s) in the denominatorIf the denominator still contains a radical, multiply the fraction by “1” where “1” is in the form of a “special binomial radical” over itselfThe “special binomial radical” is the conjugate of the denominator (same terms – opposite sign)Complete multiplication (the denominator will contain no radical)
59 Radical EquationsAn equation is called a radical equation if it contains a variable in a radicandExamples:
60 Solving Radical Equations Isolate ONE radical on one side of the equal signRaise both sides of equation to power necessary to eliminate the isolated radicalSolve the resulting equation to find “apparent solutions”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
61 Why Check When Both Sides are Raised to an Even Power? Raising both sides of an equation to a power does not always result in equivalent equationsIf both sides of equation are raised to an odd power, then resulting equations are equivalentIf both sides of equation are raised to an even power, then resulting equations are not equivalent (“extraneous solutions” may be introduced)Raising both sides to an even power, may make a false statement true:Raising both sides to an odd power never makes a false statement true:.