Algebra II – Radicals and Complex Numbers | Mathematics

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Study GuideAlgebra II–Radicals and Complex Numbers1.What Are Radicals?Radicals are an important idea in algebra. In this chapter, you will learn how radicals work and how toperform operations with them.We will explore several keytopics:•Simplifying radicals•Doing arithmetic with radicals•Simplifying rational exponents•Working with complex numbersEach of these topics helps you understand how radicals behave and how they connect to otheralgebra concepts.Understanding RadicalsRadicals can be thought of as theopposite (or undoing) of exponents.For example:•(22= 4)This means that2 squared equals 4.So if we work backwards and ask,“What number squared gives 4?”, the answer is2.This is exactly what aradicaldoes—it finds the number that produces a given result when raised to apower.The Radical SymbolThe radical symbol√is used to represent a root of a number."The radical symbol √ is used to represent a root of a number."

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Study GuideWhen there isno number written in the small corner of the radical sign, it means we are findingthesquare rootof the number inside.For example:•√4 = 2This meansthe square root of 4 is 2, because (22= 4).Other Types of RootsSometimes you will see a small number written in the corner of the radical symbol. This number tellsuswhich root we are finding.For example, a3 in the cornermeans we are finding thecube root."Sometimes you will see a small number written in the corner of the radical symbol."For example:•³√8 = 2This meansthe cube root of 8 is 2, because (23= 8).In general:•No number in the radical →square root•Small number (n) in the radical →nth rootWorking With Radical ExpressionsIn this chapter, you will learn theterms and rules used when working with radicals.Radical expressions can be treated similarly to algebraic expressions. When doing calculations, theradical part often behaves like avariable.This allows us to:•Add radicals•Subtract radicals•Multiply radicals

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Study Guide•Simplify radical expressionsLearning these operations will help you solve more complex algebra problems.Radicals and Complex NumbersRadicals are closely connected tocomplex numbers.Sometimes, we try to take the square root of anegative number. In ordinary numbers, this is notpossible. However, mathematics introduces a special number to handle this situation.This number is called theimaginary unit, written asi.The imaginary unit is defined as:This meansi represents the square root of −1.Example with Negative RadicalsLet’s look at an example:This can be rewritten as:Since:•√9 = 3•√−1 = iThe simplified result is:

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Study Guide"Let’s look at an example: √−9."2.RadicalsAradicalexpressionis written using the radical symbol. It represents a root of a number.The general form of a radical expression is:“The general form of a radical expression is √ⁿa.”Parts of a Radical ExpressionA radical expression has severalimportant parts:•Radical sign (√)–the symbol used to indicate a root.•Radicand–the number or expression inside the radical.•Index (n)–the small number that tells which root we are finding.Ifno index is written, it is understood to be2, which means we are finding thesquare root.So,is read as“the nth root of a.”Important Property of RadicalsRadicals and exponents are closely connected.Ifthen

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Study GuideThis means that raising the root to the power of its index gives back the original number.Example 1Simplify Each ExpressionLet’s practice simplifying some radical expressions.1. Square Root ExampleSince (52= 25),“1.Square Root Example”2. Cube Root ExampleBecause (43= 64),3. Fifth Root ExampleSince

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Study Guidethe answer is4. Cube Root of a Negative NumberSincethe simplified answer is5. Square Root of a Negative NumberIf we letthen

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Study GuideButnoreal number squared equals −4. Therefore:isnot a real number.Understanding Positive and Negative RootsWhen solving equations involving square roots, a number may havetwo possible values.For example:IfthenHowever, when we write theradical symbol by itself:wealways take the positive value.This is called theprincipal root.So:butAdditional ExamplesCube Root

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Study GuideIfthenandFifth Root ExampleIfthenandCube Root with Negative ValueIf

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Study GuidethensoTrueStatements About Radical ExpressionsSome general rules help us understand radicals.Even Index RootsWhen theindex is even(like square root or fourth root):•Positive numbers → real answers•Negative numbers →not real numbersExample:butisnot a real number.Odd Index RootsWhen theindex is odd(like cube root):•Positive numbers → positive answers•Negative numbers → negative answers

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Study Guide•Zero → zeroExamples:Example 2Simplifying (√x2)Let’s simplify the expression:At firstglance, it might seem that:However, this isnot always correct. The reason is thatwe do not know whether (x) is positive ornegative.For example:This shows that the square rootalways gives a nonnegative result.Because of this rule, we write:Theabsolute valueensures that the answer is always nonnegative.“Let’s simplify the expression √x².”

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