Understanding Laws of Exponents and Their Applications
Laws of exponents are essential rules for manipulating expressions with variables and exponents having the same base. These laws allow for simplification and manipulation of expressions by following specific rules regarding exponents and bases. Various examples and properties demonstrate the practical application of these laws, aiding in solving mathematical problems efficiently.
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Laws of Exponents Whenever we have variables which contain exponents and have equal bases, we can do certain mathematical operations to them. Those operations are called the Laws of Exponents bx b = base x = exponent
Laws of Exponents = ( ) xy m + = m n m n m m . 1 . 2 x x x x y m ( ) x m x x n = = m mn . 3 . 4 x m y y m x = m n 5 . , a if m n then x n x m 1 x = 5 . , b if n m then n n m x x
Other Properties of Exponents 1 0= 1 x 1= x x Any single number or variable is always to the first power ( )1 = = = = 1 1 1 1 3 3 2 2 2 a a x x x
Basic Examples = +3 = 2x 3 2x x 5x ( ) = x 3 3 4 = 4x 12 x ( ) = xy 3y 3 3 x
Basic Examples 3 3 x x = 3 y y 7 7 4 x x = = 3x 4 1 x x 5 1 x 1 = = 7 5 7 2 x x
More Examples = 14a ) 2 8 r = 2 m ( ) +4 2 = 3 37 r 4 7 2 r 7 = 2 ( 2 a ( a 5 a 8 + + = = 2 3 2 3 2 7 5 2 80r r ) = 3 12 3 2 3 5 3 3 6 15 6 15 2 5 8 m n n m n m n ( ) = = 33 3 3 3 3 3 27 x y x y 3xy 2 2 2 2 2 a 2 4 a a = = 2 2 2 3 x 9 3 x 2 9 b 4 b b 4 1 8 8 = = 3 4x 2 9 1 1 x 3 z 3 x 1 = = = 3x 3 5 2 5 2 3 z 3 z
More Examples ( ) + + + = = 4 3 3 3 2 2 3 1 2 1 1 2 21 3 7 3 7 x y z x y z xyz x y z ( )= ( ) + + + + = 2 3 1 1 1 2 1 3 3 6 8 3 2 8 3 2 48 xy xy xy x y x y ( )( )= ( ) ( ) = 2 2 1 2 2 2 3 2 1 2 1 2 2 2 2 3 2 3 2 x y x y 3 2 x y xy ( ) + + = = 4 6 2 4 4 2 6 4 6 10 9 4 9 4 36 x y x y x y x y 3 3 9 3 9 3 1 3 3 3 1 3 9 3 6 3 5 125 5 125 125 a b a a b a b a 5 a b = = = = = 3 3 6 6 3 1 3 1 3 2 3 3 6 3 2 3 27 3 27 27 a b b a b a b b 3 ab
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