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Page 1 Formulas for Exponent and Radicals Algebraic Rules for Manipulating Exponential and Radicals Expressions. In the following, n;m;k;j are arbitrary  . they can be integers or rationals or real numbers. b n b m b k =b n+mk Add exponents in the numerator and Subtract exponent in denominator. a n b m c k j = a nj b mj c kj The exponent outside the parentheses Multiplies the exponents inside. a n b m 1 = b m a n Negative exponent " ips" a fraction. b 0 = 1 b =b 1 Don't forget these Convert Radicals to Exponent notation p a =a 1=2 m p a =a 1=m m p a n =a n=m Radicals  Reducing p a 2 b =a p b Remove squares from inside m p a m b =a m p b Exponent and Radicals  Solving Equations Solve a power by a root x n=m =y,x =y m=n Solve a root by a power 1 Page 2 Formulas for Exponent and Radicals Algebraic Rules for Manipulating Exponential and Radicals Expressions. In the following, n;m;k;j are arbitrary  . they can be integers or rationals or real numbers. b n b m b k =b n+mk Add exponents in the numerator and Subtract exponent in denominator. a n b m c k j = a nj b mj c kj The exponent outside the parentheses Multiplies the exponents inside. a n b m 1 = b m a n Negative exponent " ips" a fraction. b 0 = 1 b =b 1 Don't forget these Convert Radicals to Exponent notation p a =a 1=2 m p a =a 1=m m p a n =a n=m Radicals  Reducing p a 2 b =a p b Remove squares from inside m p a m b =a m p b Exponent and Radicals  Solving Equations Solve a power by a root x n=m =y,x =y m=n Solve a root by a power 1 Example a) Simplify 2 5 3 Method 2 5 3 = 2 3 5 3 = 2 2 2 5 5 5 = 8 125 b) Simplify 2 3 2 5 3 2 Method 2 3 2 5 3 2 = 2 2 3 22 5 32 = 4 81 15; 625 = 324 15;625 Illustration: where is the negative? c) Simplify 3 4 ( the 'negative' is inside the parentheses) Method 3 4 = (3) (3) (3) (3) =81 d) Simplify 3 4 ( the 'negative' is outside the parentheses) Method 3 4 =(3) (3) (3) (3) =81 e) Simplify 2 5 3 ( the 'negative' is in the exponent) Method 2 5 3 = 1 (2=5) 3 = (or = 5 3 3 ) = 5 3 2 3 = 125 8 2 Page 3 Formulas for Exponent and Radicals Algebraic Rules for Manipulating Exponential and Radicals Expressions. In the following, n;m;k;j are arbitrary  . they can be integers or rationals or real numbers. b n b m b k =b n+mk Add exponents in the numerator and Subtract exponent in denominator. a n b m c k j = a nj b mj c kj The exponent outside the parentheses Multiplies the exponents inside. a n b m 1 = b m a n Negative exponent " ips" a fraction. b 0 = 1 b =b 1 Don't forget these Convert Radicals to Exponent notation p a =a 1=2 m p a =a 1=m m p a n =a n=m Radicals  Reducing p a 2 b =a p b Remove squares from inside m p a m b =a m p b Exponent and Radicals  Solving Equations Solve a power by a root x n=m =y,x =y m=n Solve a root by a power 1 Example a) Simplify 2 5 3 Method 2 5 3 = 2 3 5 3 = 2 2 2 5 5 5 = 8 125 b) Simplify 2 3 2 5 3 2 Method 2 3 2 5 3 2 = 2 2 3 22 5 32 = 4 81 15; 625 = 324 15;625 Illustration: where is the negative? c) Simplify 3 4 ( the 'negative' is inside the parentheses) Method 3 4 = (3) (3) (3) (3) =81 d) Simplify 3 4 ( the 'negative' is outside the parentheses) Method 3 4 =(3) (3) (3) (3) =81 e) Simplify 2 5 3 ( the 'negative' is in the exponent) Method 2 5 3 = 1 (2=5) 3 = (or = 5 3 3 ) = 5 3 2 3 = 125 8 2 More Examples f) Simplify x 3 x 7 x 5 Method x 3 x 7 x 5 =x 3+75 =x 5 g) Simplify (2a 3 b 2 )(3ab 4 ) 3 Method (2a 3 b 2 )(3ab 4 ) 3 = 2a 3 b 2 3 3 a 3 b 43 = (2 27)(a 3+3 )(b 2+12 ) = 54a 6 b 14 h) Simplify x y 3 y 2 x z 4 (give answer with only positive exponents ) Method x y 3 y 2 x z 4 = x 3 y 3 y 24 x 4 z 4 = x 3+4 y 83 z 4 = x 7 y 5 z 4 3 Page 4 Formulas for Exponent and Radicals Algebraic Rules for Manipulating Exponential and Radicals Expressions. In the following, n;m;k;j are arbitrary  . they can be integers or rationals or real numbers. b n b m b k =b n+mk Add exponents in the numerator and Subtract exponent in denominator. a n b m c k j = a nj b mj c kj The exponent outside the parentheses Multiplies the exponents inside. a n b m 1 = b m a n Negative exponent " ips" a fraction. b 0 = 1 b =b 1 Don't forget these Convert Radicals to Exponent notation p a =a 1=2 m p a =a 1=m m p a n =a n=m Radicals  Reducing p a 2 b =a p b Remove squares from inside m p a m b =a m p b Exponent and Radicals  Solving Equations Solve a power by a root x n=m =y,x =y m=n Solve a root by a power 1 Example a) Simplify 2 5 3 Method 2 5 3 = 2 3 5 3 = 2 2 2 5 5 5 = 8 125 b) Simplify 2 3 2 5 3 2 Method 2 3 2 5 3 2 = 2 2 3 22 5 32 = 4 81 15; 625 = 324 15;625 Illustration: where is the negative? c) Simplify 3 4 ( the 'negative' is inside the parentheses) Method 3 4 = (3) (3) (3) (3) =81 d) Simplify 3 4 ( the 'negative' is outside the parentheses) Method 3 4 =(3) (3) (3) (3) =81 e) Simplify 2 5 3 ( the 'negative' is in the exponent) Method 2 5 3 = 1 (2=5) 3 = (or = 5 3 3 ) = 5 3 2 3 = 125 8 2 More Examples f) Simplify x 3 x 7 x 5 Method x 3 x 7 x 5 =x 3+75 =x 5 g) Simplify (2a 3 b 2 )(3ab 4 ) 3 Method (2a 3 b 2 )(3ab 4 ) 3 = 2a 3 b 2 3 3 a 3 b 43 = (2 27)(a 3+3 )(b 2+12 ) = 54a 6 b 14 h) Simplify x y 3 y 2 x z 4 (give answer with only positive exponents ) Method x y 3 y 2 x z 4 = x 3 y 3 y 24 x 4 z 4 = x 3+4 y 83 z 4 = x 7 y 5 z 4 3 More Examples with negatives i) Simplify 6st 4 2s 2 t 2 (give answer with only positive exponents ) Negative exponents ip location: A negative exponent in the numerator moves to the denominator. And a negative exponent in the denominator moves to the numerator. Method 6st 4 2s 2 t 2 = 6ss 2 2t 4 t 2 = 3s 3 t 6 j) Simplify y 3z 3 2 (give answer with only positive exponents ) A Negative exponent ' ips' the fraction. Method y 3z 3 2 = 3z 3 y 2 = 9z 6 y 2 4 Page 5 Formulas for Exponent and Radicals Algebraic Rules for Manipulating Exponential and Radicals Expressions. In the following, n;m;k;j are arbitrary  . they can be integers or rationals or real numbers. b n b m b k =b n+mk Add exponents in the numerator and Subtract exponent in denominator. a n b m c k j = a nj b mj c kj The exponent outside the parentheses Multiplies the exponents inside. a n b m 1 = b m a n Negative exponent " ips" a fraction. b 0 = 1 b =b 1 Don't forget these Convert Radicals to Exponent notation p a =a 1=2 m p a =a 1=m m p a n =a n=m Radicals  Reducing p a 2 b =a p b Remove squares from inside m p a m b =a m p b Exponent and Radicals  Solving Equations Solve a power by a root x n=m =y,x =y m=n Solve a root by a power 1 Example a) Simplify 2 5 3 Method 2 5 3 = 2 3 5 3 = 2 2 2 5 5 5 = 8 125 b) Simplify 2 3 2 5 3 2 Method 2 3 2 5 3 2 = 2 2 3 22 5 32 = 4 81 15; 625 = 324 15;625 Illustration: where is the negative? c) Simplify 3 4 ( the 'negative' is inside the parentheses) Method 3 4 = (3) (3) (3) (3) =81 d) Simplify 3 4 ( the 'negative' is outside the parentheses) Method 3 4 =(3) (3) (3) (3) =81 e) Simplify 2 5 3 ( the 'negative' is in the exponent) Method 2 5 3 = 1 (2=5) 3 = (or = 5 3 3 ) = 5 3 2 3 = 125 8 2 More Examples f) Simplify x 3 x 7 x 5 Method x 3 x 7 x 5 =x 3+75 =x 5 g) Simplify (2a 3 b 2 )(3ab 4 ) 3 Method (2a 3 b 2 )(3ab 4 ) 3 = 2a 3 b 2 3 3 a 3 b 43 = (2 27)(a 3+3 )(b 2+12 ) = 54a 6 b 14 h) Simplify x y 3 y 2 x z 4 (give answer with only positive exponents ) Method x y 3 y 2 x z 4 = x 3 y 3 y 24 x 4 z 4 = x 3+4 y 83 z 4 = x 7 y 5 z 4 3 More Examples with negatives i) Simplify 6st 4 2s 2 t 2 (give answer with only positive exponents ) Negative exponents ip location: A negative exponent in the numerator moves to the denominator. And a negative exponent in the denominator moves to the numerator. Method 6st 4 2s 2 t 2 = 6ss 2 2t 4 t 2 = 3s 3 t 6 j) Simplify y 3z 3 2 (give answer with only positive exponents ) A Negative exponent ' ips' the fraction. Method y 3z 3 2 = 3z 3 y 2 = 9z 6 y 2 4 More Examples k) Simplify (2x 3 ) 2 (3x 4 ) (x 3 ) 4 Method (2x 3 ) 2 (3x 4 ) (x 3 ) 4 = 2 2 x 32 3x 4 x 34 = (4 3)x 6 x 4 x 12 = 12x 6+412 = 12 x 2 5Read More
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