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The document discusses exponents and rules for exponents. It defines exponents as representing the quantity A multiplied by itself N times, written as AN. It provides examples of evaluating exponents like 43. It then presents rules for exponents, including the multiplication rule ANAK = AN+K, division rule AN/AK = AN-K, power rule (AN)K = ANK, 0-power rule A0 = 1, and negative power rule A-K = 1/AK.

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Exponents
ExponentsWe write the quantity A multiplied to itself N times as AN,
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentbase
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 base
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 base
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2base
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) base
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2base
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 base
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy)base
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)base
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)base
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+K
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+KExample B.a. 5354
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+KExample B.a. 5354 = (5*5*5)(5*5*5*5)
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+KExample B.a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57b. x5y7x4y6
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+KExample B.a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57b. x5y7x4y6 = x5x4y7y6
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+KExample B.a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57b. x5y7x4y6 = x5x4y7y6 = x9y13
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+KExample B.a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57b. x5y7x4y6 = x5x4y7y6 = x9y13AN= AN – KDivision Rule:AK
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+KExample B.a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57b. x5y7x4y6 = x5x4y7y6 = x9y13AN= AN – KDivision Rule:AK56Example C. 52
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+KExample B.a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57b. x5y7x4y6 = x5x4y7y6 = x9y13AN= AN – KDivision Rule:AK56(5)(5)(5)(5)(5)(5)Example C. =52(5)(5)
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+KExample B.a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57b. x5y7x4y6 = x5x4y7y6 = x9y13AN= AN – KDivision Rule:AK56(5)(5)(5)(5)(5)(5)Example C. =52(5)(5)
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+KExample B.a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57b. x5y7x4y6 = x5x4y7y6 = x9y13AN= AN – KDivision Rule:AK56(5)(5)(5)(5)(5)(5)= 56 – 2 = 54Example C. =52(5)(5)
ExponentsPower Rule: (AN)K = ANK
ExponentsPower Rule: (AN)K = ANKExample D. (34)5
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 A1
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1A1
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0A1
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = AKAK
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – KAKAK
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AK
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 1
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 1b. 3–2
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 11b. 3–2 = 32
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 111b. 3–2 = =932
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 111b. 3–2 = =9322c. ( )–15
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 111b. 3–2 = =93221c. ( )–1==52/5
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 111b. 3–2 = =9322155c. ( )–11*===52/522
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 111b. 3–2 = =9322155c. ( )–11*===52/522abIn general ( )–K = ( )Kba
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 111b. 3–2 = =9322155c. ( )–11*===52/522abIn general ( )–K = ( )Kba2d. ( )–25
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 111b. 3–2 = =9322155c. ( )–11*===52/522abIn general ( )–K = ( )Kba25d. ( )–2= ( )252
ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 111b. 3–2 = =9322155c. ( )–11*===52/522abIn general ( )–K = ( )Kba2255d. ( )–2= ( )2=542
Exponentse. 3–1 – 40 * 2–2 =
Exponents1e. 3–1 – 40 * 2–2 =3
Exponents1– 1*e. 3–1 – 40 * 2–2 =3
Exponents11– 1*e. 3–1 – 40 * 2–2 =322
Exponents11111– 1*– =e. 3–1 – 40 * 2–2 ==3223412
Exponents11111– 1*– =e. 3–1 – 40 * 2–2 ==3223412Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents.
Exponents11111– 1*– =e. 3–1 – 40 * 2–2 ==3223412Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.
Exponents11111– 1*– =e. 3–1 – 40 * 2–2 ==3223412Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.Example E. Simplify 3–2 x4 y–6 x–8 y 23
Exponents11111– 1*– =e. 3–1 – 40 * 2–2 ==3223412Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.Example E. Simplify 3–2 x4 y–6 x–8 y 233–2 x4y–6x–8y23
Exponents11111– 1*– =e. 3–1 – 40 * 2–2 ==3223412Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.Example E. Simplify 3–2 x4 y–6 x–8 y 233–2 x4y–6x–8y23= 3–2 x4 x–8 y–6y23
Exponents11111– 1*– =e. 3–1 – 40 * 2–2 ==3223412Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.Example E. Simplify 3–2 x4 y–6 x–8 y 233–2 x4y–6x–8y23= 3–2 x4 x–8 y–6y231= x4 – 8y–6+239
Exponents11111– 1*– =e. 3–1 – 40 * 2–2 ==3223412Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.Example E. Simplify 3–2 x4 y–6 x–8 y 233–2 x4y–6x–8y23= 3–2 x4 x–8 y–6y231= x4 – 8y–6+23= x–4y17919
Exponents11111– 1*– =e. 3–1 – 40 * 2–2 ==3223412Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.Example E. Simplify 3–2 x4 y–6 x–8 y 233–2 x4y–6x–8y23= 3–2 x4 x–8 y–6y231= x4 – 8y–6+23= x–4y17= y1791919x4
Exponents11111– 1*– =e. 3–1 – 40 * 2–2 ==3223412Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.Example E. Simplify 3–2 x4 y–6 x–8 y 233–2 x4y–6x–8y23= 3–2 x4 x–8 y–6y231= x4 – 8y–6+23= x–4y17= y17=91919x4y179x4
Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3
Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8 26x–3
Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3
Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5
Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5 111==23x5*8x5
Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5 111==23x5*8x5(3x–2y3)–2 x2Example G. Simplify3–5x–3(y–1x2)3
Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5 111==23x5*8x5(3x–2y3)–2 x2Example G. Simplify3–5x–3(y–1x2)3 (3x–2y3)–2 x23–5x–3(y–1x2)3
Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5 111==23x5*8x5(3x–2y3)–2 x2Example G. Simplify3–5x–3(y–1x2)3 (3x–2y3)–2 x23–2x4y–6x2=3–5x–3y–3 x6 3–5x–3(y–1x2)3
Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5 111==23x5*8x5(3x–2y3)–2 x2Example G. Simplify3–5x–3(y–1x2)3 (3x–2y3)–2 x23–2x4y–6x23–2x4x2y–6==3–5x–3y–3 x6 3–5x–3(y–1x2)3 3–5x–3x6y–3
Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5 111==23x5*8x5(3x–2y3)–2 x2Example G. Simplify3–5x–3(y–1x2)3 (3x–2y3)–2 x23–2x4y–6x23–2x4x2y–6==3–5x–3y–3 x6 3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x6y–6=3–5x3y–3
Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5 111==23x5*8x5(3x–2y3)–2 x2Example G. Simplify3–5x–3(y–1x2)3 (3x–2y3)–2 x23–2x4y–6x23–2x4x2y–6==3–5x–3y–3 x6 3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x6y–6==3–2 – (–5) x6 – 3 y–6 – (–3)3–5x3y–3
Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5 111==23x5*8x5(3x–2y3)–2 x2Example G. Simplify3–5x–3(y–1x2)3 (3x–2y3)–2 x23–2x4y–6x23–2x4x2y–6==3–5x–3y–3 x6 3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x6y–6==3–2 – (–5) x6 – 3 y–6 – (–3)3–5x3y–3 =33 x3 y–3 =
Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5 111==23x5*8x5(3x–2y3)–2 x2Example G. Simplify3–5x–3(y–1x2)3 (3x–2y3)–2 x23–2x4y–6x23–2x4x2y–6==3–5x–3y–3 x6 3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x6y–6==3–2 – (–5) x6 – 3 y–6 – (–3)3–5x3y–3 27x3=33 x3 y–3 =y3
Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5 111==23x5*8x5(3x–2y3)–2 x2Example G. Simplify3–5x–3(y–1x2)3 (3x–2y3)–2 x23–2x4y–6x23–2x4x2y–6==3–5x–3y–3 x6 3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x6y–6==3–2 – (–5) x6 – 3y–6 – (–3)3–5x3y–3 27x3=33 x3 y–3 =y3
4 1exponents
4 1exponents

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4 1exponents

  • 2. ExponentsWe write the quantity A multiplied to itself N times as AN,
  • 3. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
  • 4. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentbase
  • 5. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 base
  • 6. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 base
  • 7. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2base
  • 8. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) base
  • 9. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2base
  • 10. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 base
  • 11. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy)base
  • 12. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)base
  • 13. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)base
  • 14. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents
  • 15. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+K
  • 16. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+KExample B.a. 5354
  • 17. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+KExample B.a. 5354 = (5*5*5)(5*5*5*5)
  • 18. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+KExample B.a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57b. x5y7x4y6
  • 19. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+KExample B.a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57b. x5y7x4y6 = x5x4y7y6
  • 20. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+KExample B.a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57b. x5y7x4y6 = x5x4y7y6 = x9y13
  • 21. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+KExample B.a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57b. x5y7x4y6 = x5x4y7y6 = x9y13AN= AN – KDivision Rule:AK
  • 22. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+KExample B.a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57b. x5y7x4y6 = x5x4y7y6 = x9y13AN= AN – KDivision Rule:AK56Example C. 52
  • 23. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+KExample B.a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57b. x5y7x4y6 = x5x4y7y6 = x9y13AN= AN – KDivision Rule:AK56(5)(5)(5)(5)(5)(5)Example C. =52(5)(5)
  • 24. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+KExample B.a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57b. x5y7x4y6 = x5x4y7y6 = x9y13AN= AN – KDivision Rule:AK56(5)(5)(5)(5)(5)(5)Example C. =52(5)(5)
  • 25. ExponentsWe write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = ANexponentExample A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2xy2 = (x)(yy) –x2 = –(xx)baseRules of Exponents Multiplication Rule: ANAK =AN+KExample B.a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57b. x5y7x4y6 = x5x4y7y6 = x9y13AN= AN – KDivision Rule:AK56(5)(5)(5)(5)(5)(5)= 56 – 2 = 54Example C. =52(5)(5)
  • 27. ExponentsPower Rule: (AN)K = ANKExample D. (34)5
  • 28. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)
  • 29. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4
  • 30. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320
  • 31. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 A1
  • 32. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1A1
  • 33. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0A1
  • 34. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
  • 35. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0
  • 36. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = AKAK
  • 37. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – KAKAK
  • 38. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK
  • 39. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AK
  • 40. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30
  • 41. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 1
  • 42. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 1b. 3–2
  • 43. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 11b. 3–2 = 32
  • 44. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 111b. 3–2 = =932
  • 45. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 111b. 3–2 = =9322c. ( )–15
  • 46. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 111b. 3–2 = =93221c. ( )–1==52/5
  • 47. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 111b. 3–2 = =9322155c. ( )–11*===52/522
  • 48. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 111b. 3–2 = =9322155c. ( )–11*===52/522abIn general ( )–K = ( )Kba
  • 49. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 111b. 3–2 = =9322155c. ( )–11*===52/522abIn general ( )–K = ( )Kba2d. ( )–25
  • 50. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 111b. 3–2 = =9322155c. ( )–11*===52/522abIn general ( )–K = ( )Kba25d. ( )–2= ( )252
  • 51. ExponentsPower Rule: (AN)K = ANKExample D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320A1Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A10-Power Rule: A0 = 1, A = 0 1A0Since = = A0 – K = A–K, we get the negative-power Rule.AKAK1Negative-Power Rule: A–K = , A = 0 AKExample D. Simplify a. 30 = 111b. 3–2 = =9322155c. ( )–11*===52/522abIn general ( )–K = ( )Kba2255d. ( )–2= ( )2=542
  • 52. Exponentse. 3–1 – 40 * 2–2 =
  • 53. Exponents1e. 3–1 – 40 * 2–2 =3
  • 54. Exponents1– 1*e. 3–1 – 40 * 2–2 =3
  • 55. Exponents11– 1*e. 3–1 – 40 * 2–2 =322
  • 56. Exponents11111– 1*– =e. 3–1 – 40 * 2–2 ==3223412
  • 57. Exponents11111– 1*– =e. 3–1 – 40 * 2–2 ==3223412Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents.
  • 58. Exponents11111– 1*– =e. 3–1 – 40 * 2–2 ==3223412Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.
  • 59. Exponents11111– 1*– =e. 3–1 – 40 * 2–2 ==3223412Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.Example E. Simplify 3–2 x4 y–6 x–8 y 23
  • 60. Exponents11111– 1*– =e. 3–1 – 40 * 2–2 ==3223412Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.Example E. Simplify 3–2 x4 y–6 x–8 y 233–2 x4y–6x–8y23
  • 61. Exponents11111– 1*– =e. 3–1 – 40 * 2–2 ==3223412Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.Example E. Simplify 3–2 x4 y–6 x–8 y 233–2 x4y–6x–8y23= 3–2 x4 x–8 y–6y23
  • 62. Exponents11111– 1*– =e. 3–1 – 40 * 2–2 ==3223412Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.Example E. Simplify 3–2 x4 y–6 x–8 y 233–2 x4y–6x–8y23= 3–2 x4 x–8 y–6y231= x4 – 8y–6+239
  • 63. Exponents11111– 1*– =e. 3–1 – 40 * 2–2 ==3223412Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.Example E. Simplify 3–2 x4 y–6 x–8 y 233–2 x4y–6x–8y23= 3–2 x4 x–8 y–6y231= x4 – 8y–6+23= x–4y17919
  • 64. Exponents11111– 1*– =e. 3–1 – 40 * 2–2 ==3223412Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.Example E. Simplify 3–2 x4 y–6 x–8 y 233–2 x4y–6x–8y23= 3–2 x4 x–8 y–6y231= x4 – 8y–6+23= x–4y17= y1791919x4
  • 65. Exponents11111– 1*– =e. 3–1 – 40 * 2–2 ==3223412Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.Example E. Simplify 3–2 x4 y–6 x–8 y 233–2 x4y–6x–8y23= 3–2 x4 x–8 y–6y231= x4 – 8y–6+23= x–4y17= y17=91919x4y179x4
  • 66. Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3
  • 67. Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8 26x–3
  • 68. Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3
  • 69. Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5
  • 70. Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5 111==23x5*8x5
  • 71. Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5 111==23x5*8x5(3x–2y3)–2 x2Example G. Simplify3–5x–3(y–1x2)3
  • 72. Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5 111==23x5*8x5(3x–2y3)–2 x2Example G. Simplify3–5x–3(y–1x2)3 (3x–2y3)–2 x23–5x–3(y–1x2)3
  • 73. Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5 111==23x5*8x5(3x–2y3)–2 x2Example G. Simplify3–5x–3(y–1x2)3 (3x–2y3)–2 x23–2x4y–6x2=3–5x–3y–3 x6 3–5x–3(y–1x2)3
  • 74. Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5 111==23x5*8x5(3x–2y3)–2 x2Example G. Simplify3–5x–3(y–1x2)3 (3x–2y3)–2 x23–2x4y–6x23–2x4x2y–6==3–5x–3y–3 x6 3–5x–3(y–1x2)3 3–5x–3x6y–3
  • 75. Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5 111==23x5*8x5(3x–2y3)–2 x2Example G. Simplify3–5x–3(y–1x2)3 (3x–2y3)–2 x23–2x4y–6x23–2x4x2y–6==3–5x–3y–3 x6 3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x6y–6=3–5x3y–3
  • 76. Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5 111==23x5*8x5(3x–2y3)–2 x2Example G. Simplify3–5x–3(y–1x2)3 (3x–2y3)–2 x23–2x4y–6x23–2x4x2y–6==3–5x–3y–3 x6 3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x6y–6==3–2 – (–5) x6 – 3 y–6 – (–3)3–5x3y–3
  • 77. Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5 111==23x5*8x5(3x–2y3)–2 x2Example G. Simplify3–5x–3(y–1x2)3 (3x–2y3)–2 x23–2x4y–6x23–2x4x2y–6==3–5x–3y–3 x6 3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x6y–6==3–2 – (–5) x6 – 3 y–6 – (–3)3–5x3y–3 =33 x3 y–3 =
  • 78. Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5 111==23x5*8x5(3x–2y3)–2 x2Example G. Simplify3–5x–3(y–1x2)3 (3x–2y3)–2 x23–2x4y–6x23–2x4x2y–6==3–5x–3y–3 x6 3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x6y–6==3–2 – (–5) x6 – 3 y–6 – (–3)3–5x3y–3 27x3=33 x3 y–3 =y3
  • 79. Exponents 23x–8 Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 26 x–3 23x–8= 23 – 6x–8–(–3 ) 26x–3= 2–3x–5 111==23x5*8x5(3x–2y3)–2 x2Example G. Simplify3–5x–3(y–1x2)3 (3x–2y3)–2 x23–2x4y–6x23–2x4x2y–6==3–5x–3y–3 x6 3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x6y–6==3–2 – (–5) x6 – 3y–6 – (–3)3–5x3y–3 27x3=33 x3 y–3 =y3