Let us try another example by combining this time

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Let us try another example by combining this time letters and numbers. Let us try to simplify 12 3 ab b . We can start by simplifying the numbers first or the letters. Let us start with the letters. 12 12 3 3 ab a b = . Now the next step is to see whether the numbers can also be simplified. 12 3 4 3 4 1 4 4 4 3 3 3 1 1 a a a a a a × × × × × × × = = = = = . Again this could have been done in one go if you are familiar with fractions. 4 12 4 3 ab a b = Using the same method as above let us try to simplify the following expressions (a) 2 a a ; (b) 5 3 a a ; (c) 5 2 x x - and (d) 2 6 2 xy x . (a) 2 1 1 a a a a a a a × × = = =
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(b) 5 5 2 3 3 1 times times a a a a a a a a a a a a a a a a a a a a a a a × × × × × × × × × = = = = × = × × × × 6447448 14243 . This expression could have also been solved in one go, as follows: 2 5 2 3 a a a / = (c) 5 3 2 1 x x x x x x x x x x x x x x x x x x x x x x x × × × × × × × × × × = = = = × × = - - - × - × - . Or 3 5 5 3 2 2 x x x x x = = - - - (d) 2 6 6 6 6 2 3 3 3 2 2 2 2 2 1 xy xy xy y y y y x x x x x x x x x × × × = = = = = = × × × × × . Or 1 2 3 6 3 2 xy y x x = . Hint: When simplifying a fraction only common factors to both the numerator and denominator are cancelled. Exercise 1.1.3 Simplify 3 5 5 2 2 9 18 ( ) ; ( ) ; ( ) ;( ) ; ( ) 3 6 a xy x x x a b c d e b y x x x - -
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11 Extra Exercises for Practice Simplify 3 5 4 3 2 3 18 36 ( ) ; ( ) ; ( ) ;( ) ; ( ) 4 6 - - xy q p y y a b c d e x q p y y 1.2 Expansion of Brackets Expanding brackets means multiplying terms to remove the brackets. Brackets expansion might in some cases involve simplification, which we have just covered. As before, we will proceed through examples. Let us try to expand the following algebraic expressions: ( 29 ( ) 2 3 2 a x y - ; ( 29 ( ) 2 3 2 b x x y + and ( 29 ( ) 3 4 1 c x - - . ( 29 ( 29 ( 29 ( ) 2 3 2 2 3 2 2 6 4 a x y x y x y - = × - × = - ( 29 ( 29 ( 29 2 ( ) 2 3 2 2 3 2 2 6 4 b x x y x x x y x xy + = × + × = + ( 29 ( 29 ( 29 ( 29 ( 29 ( ) 3 4 1 3 4 3 1 12 3 12 3 c x x x x - - = - × - - × = - - - = - + Hint: To expand an expression, with simple brackets, we multiply each term inside the brackets by the term outside the brackets. Now let us try to expand the following algebraic expressions: (2 5)(3 4) x x + - .
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12 The first thing to do is to multiply each term of the first brackets by the second brackets, i.e.: (2 5)(3 4) 2 (3 4) 5(3 4) x x x x x + - = - + - . From the double brackets our problem is reduced to expanding simple brackets. Following the same steps as above this leads us to: ( 29 ( 29 { } ( 29 ( 29 { } 2 (3 4) 5(3 4) 2 3 2 4 5 3 5 4 2 (3 4) 5(3 4) x x x x x x x x x x - + - = × - ×- + × - × - - E555555555555555F E5555555555555F The terms in the right hand side can further be expanded: ( 29 ( 29 { } ( 29 ( 29 { } { } { } 2 2 2 3 2 4 5 3 5 4 6 8 15 20 6 8 15 20 × - ×- + × - × = - + - = - + - x x x x x x x x x x Now it is time to use one of the tools you have learned so far i.e. simplification. As you can see two terms 8 x and 15 x are like terms (i.e. similar). Therefore, we need to put them together so that no term is repeated more than once in our final expression. { } 2 2 6 8 15 20 6 7 20 x x x x x - + - = + - , which is our final expression.
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