r 2 3L - 3 Zero and Negative Exponents Zero as an Exponent: For every nonzero number a, a =1. Example 1:

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1 Zero as an Exponent: For every nonzero number a, a =1. Zero and Negative Exponents Examples: r 2 3L 3 Negative Exponent: For every nonzero number a and integer n, a = Examples: I Example 1: Simplifying a Power Simplify. a. 43 b. ( l.23) Example 2: Simplifying an Exponential Expression Simplify each expression. b.i Example 3: Evaluating an Exponential Expression Evaluate 3m2t2 for m = 2 and t =

2 Multiplication Properties of Exponents Mu1tiplyin Powers With the Same Base Property: For every nonzero number a and integers m and n, am a =am. Examples: 5)_ 52TLi 51 2 Exampie 1: Multiplying Powers Rewrite each expression using each base only once. a. 4 ii ii 3 ( 5 b [1 Example 2: Multiplying Powers in an Algebraic Expression Simplify each expression. a. 2n 5 3n 2 b. 4 5x2y 3x 8 4 LN Raising a Power to a Power Property: For every nonzero number a and integers m and n, (a ) tm =atm. Examples: 2..\)3 5 ( 2 ) Example 3: Simplifying a Power Raised to a Power )6 Simplify (x3 2

3 Example 4: Simplifying an Expression With Powers Simplify c (c 3)2. Li Raising a Product to a Power Property: For every nonzero number a and b and integer n, (ab) 7 = a. Examples: (3\)a Example 5: Simplifying a Product Raised to a Power Write the expression that represents the area of the square.. 2x tl (axjz 2 Example 6: Simplifying a Product Raised to a Power 2)4. Simplify(x2)2(3xy N, chi1f,j:71 3

4 Division Properties of Exponents Dividing Powers With the Same Base Property: 3. For every nonzero number a and integers m and n, =a Example: 2 2 Example 1: Simplifying an Algebraic Expression Simplify each expression. a. 2 ft. L c1d3 r1b Cd 1 Raising a Ouotient to a Power Property: For every nonzero number a and b and integer n(j = Example: ( / Example 2: Raising a Quotient to a Power (4N3 Which expression is equivalent to I i? x) H3 (L 4

5 Example 3: Simplifying an Exponential Expression Simplify each expression. a. H 5

6 Since 52 = 25, 5 is a Since 53 = 125, 5 is a Section 7.1 Roots and Radical Expressions root of 25. root of 125. Since 54 =625,5isaC(+k Since 55 =3125,5isa rooto rootof 3j Definition of the nth Root: For any real numbers a and b, and any positive integer n, if a = b, then a is an nth root of b. 9 oc c*d O 4 rcc*s )I(hC5flOftoi khq C 4Z j Summary of the possible real roots of a real number. Number of Rea nth NLInbef of ReI,,th Type of Nimiber Roots When n s Even ROGtS Weii IF Js Odd :JciF(\) c QOch ñ Example 1: Finding All Real Roots Find all the real roots. a. The cube roots of 8, 1000, and DC t\ I 0 hq1\cc4 Iäo \ \ hq\ Q_I OO& 16 b. The fourth roots of 1, , and. L jii.t c{th rooi od VH and (r)z I V 1T) hcjl QJJ (TJ3 S 1JTD1O b 1OVt&, I 1 5

7 Radical Sign: \,cj ko \r$&cck( Radicand: [ cctr rdr* tijç oj Index: PrincipalRoot: Th. Example 2: Finding Roots JQ n c n JU)cr TYL inc 4 rc O 6 ThQ Lth rc ct Q$ a _codco \CkS T +o oc*. Find each realnumber root. a.vi ; b. J ioo S Sc)c\m c@oj 3 iu&br Example 3: Simplifying Radical Expressions Simplify each radical expression. a. b. Ja3b6 C. Jx y 5T:: jl 2

8 If and [ are real numbers, then. = Example 1: Multiplying Radicals MultipIyin Radical Expressions Property:. pap+j \ S rob a QcJ nuidg. cbqs nc* cppk scc \ Multiply. Simplify if possible. = Example 2: Simplifying Radical Expressions Simplify each expression. Assume that all variables are positive. Then absolute value symbols are never needed in the simplified expression. a. \J b. ki8on J10z Dividing Radical Expressions Property: 3 If and arerealnumbersand bo,then )ion Section 7.2 Multiplyin2 and Dividing Radical Expressions

9 Example 4: Dividing Radicals Divide and simplify. Assume that all variables are positive. a. b 3 Rationalize the Denominator f)jç \\Q áqf1ofl ()Q O \Ddcc. Example 5: Rationalizing the Denominator Rationalize the denominator of each expression. Assume that all variables are positive. a J \{3 _1çI nco1 I 4D b 5X\/ c. 3J_ V 3x c.l E L_ 4

10 5 J9Y 33 Simplify 6lu + 4J [CC Section 7.3 Binomial Radical Expressions Example 2: Simplifying Before Adding and Subtracting Like Radicals: ea\ ftss Example 3: Multiplying Binomial Radical Expressions Multiply (3+ 2J)(2 + 4J). Example!: Adig ubtingdial W &... bxiq Pci: t 4 3In. Add or subtract if possible. Ljoj a. 5k1 3& b. 4Ii+5J + L: 4l_ rcc

11 Example 4: Multiplying Conjugates Multiply (2+ J)(2 Fo IL cc QT DrQnc c \ (cbbcb2 ()z Example 5: Rationalizing Binomial Radical Denominators Rationalize the denominator of 3+ (3 I 4 5 Lj 6

12 b2 Example 1: Simplifying Expressions With Rational Exponents VL CO 1sScor. Section 7.4 Rational Exponents Simplify each expression. a. 125 Rational Exponent: b. Write the radical expressions J1 and (J)2 in exponential form. a. Write the exponential expressions x and y2.5 in radical form. Example 2: Converting to and From Radical Form J OCrdk a = and a = ( If the nth root of a is a real number and m is an integer, then Definition of Rational Exponents: TL (Th( 1o.1oo = C) b. 5 5} \) 5

13 \j Write (16y )2.. L3 (am)n =am b. a ] Lg Example 435 = a 9Vi (ab) 11 =a n11 5 1) a11 Property Example 4: Simplifying Numbers With Rational Exponents 8) in simplest form. ExampleS: Writing Expressions in Simplest Form _33/5 \j(33) 3. Simplify each number. (a a [L \3 f \11 _n m a11 L b11 ai 3 = a. ( 32) L L Ni 1 ) 1(s I \1J4/ 1 _i N I Let m and n represent rational numbers. Assume that no denominator equals 0. a am+n Summary of Properties of Rational Exponents: 4 (

14 sc tcu +hp rod cc] c Solving a Radical Equation. or Sc& 0 c Radical Equation: ç rn \Jc QE* 53DQ \) 9 50 : 50 ( 50 x...(es I x A V. I. 1 F cnc 50 (x x :: 50 r 3 3Q tb 5Q. X 53 CbQC. 0 cj9 Solve 2(x 2) =50. Example 2: Solving Radical Equations With Rational Exponents i CHY +i4 \3K H Solve 2+iJ3x 2=6. Exampie 1: Solving Square Root Equations Qa\O c+ + cjxthc thsq bow c$ Section 7.5 Solve Square Root and Other Radical Equations

15 Extraneous Solutions: A tuecn d cn c± cn i Qd om Example 3: Checking For Extraneous Solutions Solve Jx 3 +5 = x. Check for extraneous solutions. 5K 5 (7S (i Xx Al) 43 A 32jj f5 _o \rte 31 J ,/ \J H Example 4: Solving Equations With Two Rational Exponents Solve (2x + 1)0.5 (3x + 4) Check for extraneous solutions. (3KH) c (x i)a (±i) z(*t) q4q+( 3x4 4xx 3 D(S o1., So I q ChQch (./) + O05 /3/. 17 \. (%) /5iEz L (i:) (E:) Q3 (3(/) /c f \Q.5 vi) 1 L / I) ) oc.7 (3 1 2 ±0 0 0 (4K3jy o 10 rai ium.br