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_ _ Zero and Negative Exponents Zero as an Exponent: For every nonzero number a, a° =1. Examples: r° 2 3L -‘ Negative Exponent: For every nonzero number a and integer n, a = I Examples: 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 3m t for m 2 = 2 and t = -3. 37 12 3 Multiplication Properties of Exponents Mu1tiplyin Powers With the Same Base Property: For every nonzero number a and integers m and n, am a” Examples: 5)_ 5 2 TLi = 51 . m a 2’ Exampie 1: Multiplying Powers Rewrite each expression using each base only once. a. ii 4 •ii 3 b. ( 52 .52 5 ‘ 2 2 - [1 Example 2: Multiplying Powers in an Algebraic Expression Simplify each expression. a. 5 3n 2n 2 b. 5x2y 4 •3x 8 4 °LN Raising a Power to a Power Property: For every nonzero number a and integers m and n, (a )” tm Examples: 2..\)3 5- = t”. a m (—2’) Example 3: Simplifying a Power Raised to a Power Simplify 3 (x )6 2 Example 4: Simplifying an Expression With Powers Simplify c’(c . 2 ) 3 Li Raising a Product to a Power Property: For every nonzero number a and b and integer n, (ab)’ 7 = (3\)a. Examples: 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 (3xy (x . 4 ) Simplify 2 N, cHI1f ,J:71 3 a’. 32. 2 92 Division Properties of Exponents Dividing Powers With the Same Base Property: For every nonzero number a and integers m and n, Example: = . 3 a’ 2 2 Example 1: Simplifying an Algebraic Expression Simplify each expression. 3 d 1 c -2 a. ft. L 1 Cd 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) 3 H b 1 r (L - 4 Example 3: Simplifying an Exponential Expression Simplify each expression. a. H— — — 5 ___ _________ ____________ ______________ Section 7.1 Since 52 Since 53 Since 54 Since 55 = 25, 5 is a = 125, 5 is a Roots and Radical Expressions root of 25. root of 125. rooto =625,5isaC(+k rootof =3125,5isa 3j Definition of the nth Root: For any real numbers a and b, and any positive integer n, if a’ root of b. oc 9 - = c*d b, then a is an nth — O 4- rcc*s )-I(hC5flOftoi - 4Z khQ C ‘ j Summary of the possible real roots of a real number. Type of Nimiber Number of Rea’ nth Roots When n s Even NLInbef’ of ReI ,,th ROGtS Weii IF Js Odd -:JciF(\) c QOc h ñ Example 1: Finding All Real Roots Find all the real roots. a. The cube roots of 8, -1000, and I hQ1\cc4 t\ -DC \‘\ hQ\’Q_I OO& b. The fourth roots of 1, -0.0001, and L ‘jiI.t 0 Iäo 16 —. c{th rooi- od V 1T) hCJL QJJ -- , (TJ3 - S VH and 1JTD1O - I 1 5 b 1O (-r)Z I &t V \,cJ —ko Radical Sign: Radicand: Index: \r$&CCk( cctr rdr * tijç oJ —[- Th° PrincipalRoot: TYL a -. n c n’JU)cr -JQ i 4 rc nc ThQ Lth O rc’ct \CkS _codcO +o oc*. 6 Q$ T Example 2: Finding Roots Find each real-number root. a.Vi b. c@oJ iu&br ‘S Sc)c\m 3 -; Example 3: Simplifying Radical Expressions Simplify each radical expression. a. b. Ja3b6 C. Jx’y J—ioo - 5T:: jL 2 __ Section 7.2 Multiplyin2 and Dividing Radical Expressions MultipIyin Radical Expressions Property: If and ‘[ are real numbers, then . = Example 1: Multiplying Radicals Multiply. Simplify if possible. -\ = . \- pap+j cbQS nc* cppk S rob- a QcJ nuiDg. scc 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. b. \J kI8On z 1 J 0 )ion Dividing Radical Expressions Property: If and arerealnumbersand bO,then 3 ___ Example 4: Dividing Radicals Divide and simplify. Assume that all variables are positive. a. 3 b Rationalize the Denominator ()Q O f)J ç- \Dd cc. -\-\Q- áQf1ofl ncO1 Example 5: Rationalizing the Denominator Rationalize the denominator of each expression. Assume that all variables are positive. a •J _1çI \{3 4D ‘I b - 5X\/ — c. 3J_ V 3x - .L’E c L_ 4 Section 7.3 Like Radicals: - Binomial Radical Expressions ea\ ftSS [CC -W’&... bxiQ ubtingdial Example!: Adig Add or subtract if possible. a. 5k1—3& b. 4Ii+5J rcC- Example 2: Simplifying Before Adding and Subtracting Simplify 6-lu + 4J — 3-In. J9Y -33 4l_ -L: + Example 3: Multiplying Binomial Radical Expressions Multiply (3+ 2J)(2 + 4J). Pci : t 4 Ljoj 5 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+ 4 (3 - I - ‘-5 -Lj 6 \ Section 7.4 Rational Exponents Rational Exponent: CO - 1sScor. VL Example 1: Simplifying Expressions With Rational Exponents Simplify each expression. a. 125 b. 5 5} 1o.1oo 5 \) C) = Definition of Rational Exponents: If the nth root of a is a real number and m is an integer, then and a = a= - TL ( ‘J Example 2: Converting to and From Radical Form 5 in radical form. . 2 a. Write the exponential expressions x and y b. Write the radical expressions J1 and (J)2 in exponential form. 2 b (Th( OCrdk Summary of Properties of Rational Exponents: Let m and n represent rational numbers. Assume that no denominator equals 0. Example Property ( 4 ]‘Lg am+n 1’1 a a (am)n =am - 1 =a 1 n 11 11 (ab) a _n L3 )2.. - 5 1) 9Vi a” m = L 11 a f (a \11 “ [L’\3 a 11 b 3 ai Example 4: Simplifying Numbers With Rational Exponents Simplify each number. a. = _33/5 435 b. (—32) 3 \j(33) L . - L4 N I ExampleS: Writing Expressions in Simplest Form Write (16y ) in simplest form. 8 1(s”I J4/ 1 \ —_i-— \j Ni 1 ) 3 8 .- Section 7.5 Solve Square Root and Other Radical Equations Radical Equation: ç0 rn Solving a Radical Equation. c+- + \xI) @ ‘\Jc c sc tcU cjxthc +hp rod - cc] thSQ c boW’ QE* or Sc& c$ Qa\ O Exampie 1: Solving Square Root Equations - Solve 2+iJ3x—2=6. \3K- H CHY -i-• +i4 - Example 2: Solving Radical Equations With Rational Exponents Solve 2(x—2) =50. X- 5Q. 53 tb (x x CbQC. 0 cj9 r’3 3Q - :: 50 ‘A - V I’ -. 1 ‘F cNc .‘ 50 x-.--. .(ES x\) : I •( 9 50 50 50 53DQ Extraneous Solutions: •i Qd om A tuEcn d cn c± cn Example 3: Checking For Extraneous Solutions Solve -Jx —3 +5 = x. Check 5K for extraneous solutions. (7S -5 (i-’Xx Al) 7 \rTE ÷31 A J+ 5 f5 32jj .3 \J 7 ÷5 —H -4-3 _o 1 7 ,/ Example 4: Solving Equations With Two Rational Exponents Solve (2x + 1)0.5 — (3x + 4)0.25 (3KH) 0. Check for extraneous solutions. ChQch (./) + O05 (3(/) /c . /3/ 17 ÷ c \ (±i) L (%)° z(*t) / q q 4 +( -1- -. L . - So -, -- (i:) (E:) Q3 3x4 q ) (3 iEz 5 oc -1 4xx -3 1 o 0 -- (x i)a D(S .7 0 -- 2 f I \Q.5 v-I) rai (4K-3jy o 10 / “I) ±0 i-um.br-