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Exponents
Exponents We write the quantity A multiplied to itself N times as AN,
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent base
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 base
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 base
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2 base
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) base
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 base
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 base
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) base
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5)
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6 = x9y13
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6 = x9y13 AN = AN – K Division Rule: AK
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6 = x9y13 AN = AN – K Division Rule: AK 56 Example C. 52
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6 = x9y13 AN = AN – K Division Rule: AK 56 (5)(5)(5)(5)(5)(5) Example C. = 52 (5)(5)
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6 = x9y13 AN = AN – K Division Rule: AK 56 (5)(5)(5)(5)(5)(5) Example C. = 52 (5)(5)
Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6 = x9y13 AN = AN – K Division Rule: AK 56 (5)(5)(5)(5)(5)(5) = 56 – 2 = 54 Example C. = 52 (5)(5)
Exponents Power Rule: (AN)K = ANK
Exponents Power Rule: (AN)K = ANK Example D. (34)5
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 A1
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 A1
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0 A1
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = AK AK
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K AK AK
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1 b. 3–2
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1 1 b. 3–2 = 32
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1 1 1 b. 3–2 = = 9 32
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1 1 1 b. 3–2 = = 9 32 2 c. ( )–1 5
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1 1 1 b. 3–2 = = 9 32 2 1 c. ( )–1 = = 5 2/5
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1 1 1 b. 3–2 = = 9 32 2 1 5 5 c. ( )–1 1* = = = 5 2/5 2 2
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1 1 1 b. 3–2 = = 9 32 2 1 5 5 c. ( )–1 1* = = = 5 2/5 2 2 a b In general ( )–K = ( )K b a
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1 1 1 b. 3–2 = = 9 32 2 1 5 5 c. ( )–1 1* = = = 5 2/5 2 2 a b In general ( )–K = ( )K b a 2 d. ( )–2 5
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1 1 1 b. 3–2 = = 9 32 2 1 5 5 c. ( )–1 1* = = = 5 2/5 2 2 a b In general ( )–K = ( )K b a 2 5 d. ( )–2 = ( )2 5 2
Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1 1 1 b. 3–2 = = 9 32 2 1 5 5 c. ( )–1 1* = = = 5 2/5 2 2 a b In general ( )–K = ( )K b a 2 25 5 d. ( )–2 = ( )2 = 5 4 2
Exponents e. 3–1 – 40 * 2–2 =
Exponents 1 e. 3–1 – 40 * 2–2 = 3
Exponents 1 – 1* e. 3–1 – 40 * 2–2 = 3
Exponents 1 1 – 1* e. 3–1 – 40 * 2–2 = 3 22
Exponents 1 1 1 1 1 – 1* – = e. 3–1 – 40 * 2–2 = = 3 22 3 4 12
Exponents 1 1 1 1 1 – 1* – = e. 3–1 – 40 * 2–2 = = 3 22 3 4 12 Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents.
Exponents 1 1 1 1 1 – 1* – = e. 3–1 – 40 * 2–2 = = 3 22 3 4 12 Although 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.
Exponents 1 1 1 1 1 – 1* – = e. 3–1 – 40 * 2–2 = = 3 22 3 4 12 Although 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
Exponents 1 1 1 1 1 – 1* – = e. 3–1 – 40 * 2–2 = = 3 22 3 4 12 Although 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 3–2 x4y–6x–8y23
Exponents 1 1 1 1 1 – 1* – = e. 3–1 – 40 * 2–2 = = 3 22 3 4 12 Although 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 3–2 x4y–6x–8y23 = 3–2 x4 x–8 y–6y23
Exponents 1 1 1 1 1 – 1* – = e. 3–1 – 40 * 2–2 = = 3 22 3 4 12 Although 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 3–2 x4y–6x–8y23 = 3–2 x4 x–8 y–6y23 1 = x4 – 8y–6+23 9
Exponents 1 1 1 1 1 – 1* – = e. 3–1 – 40 * 2–2 = = 3 22 3 4 12 Although 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 3–2 x4y–6x–8y23 = 3–2 x4 x–8 y–6y23 1 = x4 – 8y–6+23 = x–4y17 9 1 9
Exponents 1 1 1 1 1 – 1* – = e. 3–1 – 40 * 2–2 = = 3 22 3 4 12 Although 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 3–2 x4y–6x–8y23 = 3–2 x4 x–8 y–6y23 1 = x4 – 8y–6+23 = x–4y17 = y17 9 1 9 1 9x4
Exponents 1 1 1 1 1 – 1* – = e. 3–1 – 40 * 2–2 = = 3 22 3 4 12 Although 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 3–2 x4y–6x–8y23 = 3–2 x4 x–8 y–6y23 1 = x4 – 8y–6+23 = x–4y17 = y17 = 9 1 9 1 9x4 y17 9x4
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 1 1 1 = = 23 x5 * 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 1 1 1 = = 23 x5 * 8x5 (3x–2y3)–2 x2 Example G. Simplify 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 1 1 1 = = 23 x5 * 8x5 (3x–2y3)–2 x2 Example G. Simplify 3–5x–3(y–1x2)3 (3x–2y3)–2 x2 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 1 1 1 = = 23 x5 * 8x5 (3x–2y3)–2 x2 Example G. Simplify 3–5x–3(y–1x2)3 (3x–2y3)–2 x2 3–2x4y–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 1 1 1 = = 23 x5 * 8x5 (3x–2y3)–2 x2 Example G. Simplify 3–5x–3(y–1x2)3 (3x–2y3)–2 x2 3–2x4y–6x2 3–2x4x2y–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 1 1 1 = = 23 x5 * 8x5 (3x–2y3)–2 x2 Example G. Simplify 3–5x–3(y–1x2)3 (3x–2y3)–2 x2 3–2x4y–6x2 3–2x4x2y–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 1 1 1 = = 23 x5 * 8x5 (3x–2y3)–2 x2 Example G. Simplify 3–5x–3(y–1x2)3 (3x–2y3)–2 x2 3–2x4y–6x2 3–2x4x2y–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 1 1 1 = = 23 x5 * 8x5 (3x–2y3)–2 x2 Example G. Simplify 3–5x–3(y–1x2)3 (3x–2y3)–2 x2 3–2x4y–6x2 3–2x4x2y–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 1 1 1 = = 23 x5 * 8x5 (3x–2y3)–2 x2 Example G. Simplify 3–5x–3(y–1x2)3 (3x–2y3)–2 x2 3–2x4y–6x2 3–2x4x2y–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 1 1 1 = = 23 x5 * 8x5 (3x–2y3)–2 x2 Example G. Simplify 3–5x–3(y–1x2)3 (3x–2y3)–2 x2 3–2x4y–6x2 3–2x4x2y–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
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4 1exponents

  • 2. Exponents We write the quantity A multiplied to itself N times as AN,
  • 3. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
  • 4. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent base
  • 5. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 base
  • 6. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 base
  • 7. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2 base
  • 8. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) base
  • 9. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 base
  • 10. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 base
  • 11. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) base
  • 12. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base
  • 13. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base
  • 14. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents
  • 15. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K
  • 16. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354
  • 17. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5)
  • 18. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6
  • 19. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6
  • 20. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6 = x9y13
  • 21. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6 = x9y13 AN = AN – K Division Rule: AK
  • 22. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6 = x9y13 AN = AN – K Division Rule: AK 56 Example C. 52
  • 23. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6 = x9y13 AN = AN – K Division Rule: AK 56 (5)(5)(5)(5)(5)(5) Example C. = 52 (5)(5)
  • 24. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6 = x9y13 AN = AN – K Division Rule: AK 56 (5)(5)(5)(5)(5)(5) Example C. = 52 (5)(5)
  • 25. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN exponent Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base Rules of Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6 = x9y13 AN = AN – K Division Rule: AK 56 (5)(5)(5)(5)(5)(5) = 56 – 2 = 54 Example C. = 52 (5)(5)
  • 26. Exponents Power Rule: (AN)K = ANK
  • 27. Exponents Power Rule: (AN)K = ANK Example D. (34)5
  • 28. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)
  • 29. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4
  • 30. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320
  • 31. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 A1
  • 32. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 A1
  • 33. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0 A1
  • 34. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1
  • 35. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0
  • 36. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = AK AK
  • 37. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K AK AK
  • 38. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK
  • 39. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK
  • 40. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30
  • 41. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1
  • 42. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1 b. 3–2
  • 43. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1 1 b. 3–2 = 32
  • 44. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1 1 1 b. 3–2 = = 9 32
  • 45. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1 1 1 b. 3–2 = = 9 32 2 c. ( )–1 5
  • 46. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1 1 1 b. 3–2 = = 9 32 2 1 c. ( )–1 = = 5 2/5
  • 47. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1 1 1 b. 3–2 = = 9 32 2 1 5 5 c. ( )–1 1* = = = 5 2/5 2 2
  • 48. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1 1 1 b. 3–2 = = 9 32 2 1 5 5 c. ( )–1 1* = = = 5 2/5 2 2 a b In general ( )–K = ( )K b a
  • 49. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1 1 1 b. 3–2 = = 9 32 2 1 5 5 c. ( )–1 1* = = = 5 2/5 2 2 a b In general ( )–K = ( )K b a 2 d. ( )–2 5
  • 50. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1 1 1 b. 3–2 = = 9 32 2 1 5 5 c. ( )–1 1* = = = 5 2/5 2 2 a b In general ( )–K = ( )K b a 2 5 d. ( )–2 = ( )2 5 2
  • 51. Exponents Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34)= 34+4+4+4+4 = 34*5 = 320 A1 Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 0-Power Rule: A0 = 1, A = 0 1 A0 Since = = A0 – K = A–K, we get the negative-power Rule. AK AK 1 Negative-Power Rule: A–K = , A = 0 AK Example D. Simplify a. 30 = 1 1 1 b. 3–2 = = 9 32 2 1 5 5 c. ( )–1 1* = = = 5 2/5 2 2 a b In general ( )–K = ( )K b a 2 25 5 d. ( )–2 = ( )2 = 5 4 2
  • 52. Exponents e. 3–1 – 40 * 2–2 =
  • 53. Exponents 1 e. 3–1 – 40 * 2–2 = 3
  • 54. Exponents 1 – 1* e. 3–1 – 40 * 2–2 = 3
  • 55. Exponents 1 1 – 1* e. 3–1 – 40 * 2–2 = 3 22
  • 56. Exponents 1 1 1 1 1 – 1* – = e. 3–1 – 40 * 2–2 = = 3 22 3 4 12
  • 57. Exponents 1 1 1 1 1 – 1* – = e. 3–1 – 40 * 2–2 = = 3 22 3 4 12 Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents.
  • 58. Exponents 1 1 1 1 1 – 1* – = e. 3–1 – 40 * 2–2 = = 3 22 3 4 12 Although 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. Exponents 1 1 1 1 1 – 1* – = e. 3–1 – 40 * 2–2 = = 3 22 3 4 12 Although 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. Exponents 1 1 1 1 1 – 1* – = e. 3–1 – 40 * 2–2 = = 3 22 3 4 12 Although 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 3–2 x4y–6x–8y23
  • 61. Exponents 1 1 1 1 1 – 1* – = e. 3–1 – 40 * 2–2 = = 3 22 3 4 12 Although 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 3–2 x4y–6x–8y23 = 3–2 x4 x–8 y–6y23
  • 62. Exponents 1 1 1 1 1 – 1* – = e. 3–1 – 40 * 2–2 = = 3 22 3 4 12 Although 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 3–2 x4y–6x–8y23 = 3–2 x4 x–8 y–6y23 1 = x4 – 8y–6+23 9
  • 63. Exponents 1 1 1 1 1 – 1* – = e. 3–1 – 40 * 2–2 = = 3 22 3 4 12 Although 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 3–2 x4y–6x–8y23 = 3–2 x4 x–8 y–6y23 1 = x4 – 8y–6+23 = x–4y17 9 1 9
  • 64. Exponents 1 1 1 1 1 – 1* – = e. 3–1 – 40 * 2–2 = = 3 22 3 4 12 Although 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 3–2 x4y–6x–8y23 = 3–2 x4 x–8 y–6y23 1 = x4 – 8y–6+23 = x–4y17 = y17 9 1 9 1 9x4
  • 65. Exponents 1 1 1 1 1 – 1* – = e. 3–1 – 40 * 2–2 = = 3 22 3 4 12 Although 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 3–2 x4y–6x–8y23 = 3–2 x4 x–8 y–6y23 1 = x4 – 8y–6+23 = x–4y17 = y17 = 9 1 9 1 9x4 y17 9x4
  • 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 1 1 1 = = 23 x5 * 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 1 1 1 = = 23 x5 * 8x5 (3x–2y3)–2 x2 Example G. Simplify 3–5x–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 1 1 1 = = 23 x5 * 8x5 (3x–2y3)–2 x2 Example G. Simplify 3–5x–3(y–1x2)3 (3x–2y3)–2 x2 3–5x–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 1 1 1 = = 23 x5 * 8x5 (3x–2y3)–2 x2 Example G. Simplify 3–5x–3(y–1x2)3 (3x–2y3)–2 x2 3–2x4y–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 1 1 1 = = 23 x5 * 8x5 (3x–2y3)–2 x2 Example G. Simplify 3–5x–3(y–1x2)3 (3x–2y3)–2 x2 3–2x4y–6x2 3–2x4x2y–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 1 1 1 = = 23 x5 * 8x5 (3x–2y3)–2 x2 Example G. Simplify 3–5x–3(y–1x2)3 (3x–2y3)–2 x2 3–2x4y–6x2 3–2x4x2y–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 1 1 1 = = 23 x5 * 8x5 (3x–2y3)–2 x2 Example G. Simplify 3–5x–3(y–1x2)3 (3x–2y3)–2 x2 3–2x4y–6x2 3–2x4x2y–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 1 1 1 = = 23 x5 * 8x5 (3x–2y3)–2 x2 Example G. Simplify 3–5x–3(y–1x2)3 (3x–2y3)–2 x2 3–2x4y–6x2 3–2x4x2y–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 1 1 1 = = 23 x5 * 8x5 (3x–2y3)–2 x2 Example G. Simplify 3–5x–3(y–1x2)3 (3x–2y3)–2 x2 3–2x4y–6x2 3–2x4x2y–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 1 1 1 = = 23 x5 * 8x5 (3x–2y3)–2 x2 Example G. Simplify 3–5x–3(y–1x2)3 (3x–2y3)–2 x2 3–2x4y–6x2 3–2x4x2y–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