Core Mathematics 1 Indices & Surds
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1 Regent College Maths Department Core Mathematics Indices & Surds
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3 Indices September 0 C Note Laws of indices for all rational exponents. The equivalence of We should already know from GCSE, the three Laws of indices : m n mn ( I) a a a (e.g. a a a ) m n mn 7 ( II) a a a (e.g. a a a ) 7 ( III) a a (e.g. a a ) m n mn m n a and n m a should be known. In addition to these we need to remember the following: REMEMBER n n n,,, n a a a a a a etc. a a n n a a, a a So, for example: n a, a n a a So, for example: 8 m m n n n m a a a So, for example, Copyright - For AS, A notes and IGCSE / GCSE worksheets 7
4 September 0 C Note REMEMBER 0 a and a a for all values of a. Exam Question Given that x = and y =, (a) find the exact value of x and the exact value of y, so. Hence x. Hence y (b) calculate the exact value of y x. yx 8 Exam Question (a) Given that 8 = k, write down the value of k. k (b) Given that x = 8 x, find the value of x. x x 8 x x 6x x 6x x 6 6 x x Copyright - For AS, A notes and IGCSE / GCSE worksheets 8
5 Surds September 0 C Note Use and manipulation of surds. Students should be able to rationalise denominators. The square roots of certain numbers are integers (e.g. 9 ) but when this is not the case it is often easier to leave the square roots sign in the expression (e.g. it is simpler to write than it is to write the value our calculator gives, i.e ). Numbers of the form,, etc. are called surds. We need to be able to simplify expressions involving surds. It is important to realise the following : ab a b a b e. g. 6. Now we see from the above that we can sometimes simplify surds. For example 7 or 0 and 7 We can only simplify an expression of the form the number was in the above example). a if a has a factor which is a perfect square ( as Multiplying surds: Multiply out brackets in usual way (using FOIL or a similar method). Then collect similar terms. e.g Copyright - For AS, A notes and IGCSE / GCSE worksheets 9
6 Dividing surds: For example simplify The denominator of this fraction is.. September 0 C Note This is irrational and we need to be able to express this in a form in which the denominator is rational. To do this we must multiply top and bottom of the fraction by an expression that will rationalise the denominator. We saw from above that so we multiply top and bottom by So we have the following 0. Rationalising the denominator: If the denominator is a b then multiply top and bottom by a b. If the denominator is a b then multiply top and bottom by a b. If the denominator is a b then multiply top and bottom by a b. If the denominator is a b then multiply top and bottom by a b. Exam Question Given that ( + 7)( 7) = a + b7, where a and b are integers, (a) find the value of a and the value of b so a, b. Given that 7 7 = c + d7 where c and d are rational numbers, (b) find the value of c and the value of d Copyright - For AS, A notes and IGCSE / GCSE worksheets 0
7 . (a) Write down the value of 6. (b) Find the value of 6. () () Jan 00, Q. (a) Write down the value of 8. (b) Find the value of 8. () () May 00, Q. (a) Write in the form a, where a is an integer. () (b) Express ( + ) ( ) in the form b + c, where b and c are integers. () Jan 006, Q. (a) Expand and simplify ( + ) ( ). () (b) Express 6 + in the form a + b, where a and b are integers. () May 006, Q6. (a) Express 08 in the form a, where a is an integer. (b) Express ( ) in the form b + c, where b and c are integers to be found. () () Jan 007, Q 6. Simplify ( + )( ). () May 007, Q
8 7. (a) Find the value of 8. () (b) Simplify x x. () May 007, Q 8. (a) Write down the value of 6. (b) Simplify (6x ). () () Jan 008, Q 9. Simplify +, giving your answer in the form a + b, where a and b are integers. () Jan 008, Q 0. (a) Write down the value of. (b) Find the value of. () () Jan 009, Q. Expand and simplify ( 7 + )( 7 ). () Jan 009, Q
9 . Simplify (a) ( 7) (b) (8 + )( ) () () June 009, Q. Given that = a, find the value of a.. (a) Expand and simplify (7 + )( ). () June 009, Q () (b) Express in the form a + b, where a and b are integers. () Jan 00, Q. Write (7) (7) in the form k x, where k and x are integers. 6. (a) Find the value of 6. () May 00, Q () (b) Simplify 7. Simplify x x., () Jan 0, Q giving your answer in the form p + q, where p and q are rational numbers. () Jan 0, Q
10 8. Find the value of (a), () (b). () May 0, Q 9. (a) Simplify + 8, giving your answer in the form a, where a is an integer. (b) Simplify + 8, + giving your answer in the form b + c, where b and c are integers. 0. (a) Evaluate ( ), giving your answer as an integer. () () Jan 0, Q () (b) Simplify fully x. () May 0, Q. Show that 8 can be written in the form a + b, where a and b are integers. () May 0, Q
11 . Express 8 x + in the form y, stating y in terms of x. () Jan 0, Q. (i) Express ( 8)( + ) in the form a + b, where a and b are integers. () (ii) Express in the form c, where c is an integer () Jan 0, Q. Simplify 7 +, giving your answer in the form a + b, where a and b are integers. () May 0, Q. (a) Find the value of 8. (x ) (b) Simplify fully x. () () May 0, Q 6. Find 7 in the form k, where k is an integer. () May 0_R, Q
12 7. Solve (a) y = 8, (b) x x + = 8. () () May 0_R, Q 8. (a) Write down the value of. (b) Simplify fully (x ). 9. (a) Evaluate 8 () () May 0, Q () (b) Simplify fully x x () May 0_R, Q
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