Maths for A Level Physics - updated edition

For example, 1 2 = 2 −1 , so 1 2 × 8 = 2 −1 × 2 3 = 2 (−1 + 3) = 2 2 = 4, which is good! In other words, our new understanding of what a −n means gives us what we know to be the right answer. What we’re now doing is letting the formal laws of indices guide us to assign meanings to a m and a n , even when m and n are not positive whole numbers – for which the laws were ϐirst written. We continue this assignment of meaning by putting n = 0 in the ϐirst two laws, giving a m × a 0 = a ( m + 0) = a m and a m a 0 = a ( m − 0) = a m So multiplying or dividing by a 0 has no effect on a m . We conclude that, for any a , a 0 = 1. 4.2.3 Fractional indices Put m = n = 1 2 into the ϐirst law. Then: a 1 2 × a 1 2 = a 1 = a But √ a is deϐined by √ a × √ a = a We conclude that: a 1 2 = √ a . And, in general, a 1 n = n √ a What about a m n ? Now ( a m n ) n = a m n × n = a m , so a m n = n √ a m . You should be able to show that this can be written ( n √ a ) m . This is left as an exercise. We can now interpret numbers with decimal indices, for example 2 0.4 . 2 0.4 = 2 410 = 10 √ 2 4 . We could equally well write this as 5 √ 2 2 . Example B: Express 1 4√ 2 as 2 x . 1 4√ 2 = 1 2 2 × 2 1 2 = 1 2 ( 2 + 1 2 ) = 1 2 5 2 = 2 − 5 2 This could equally well be written2 −2.5 . It is sensible to consider the order of the calculation when simplifying numbers with fractional indices. For example, what is 27 4 3 ? We could write this as (27 4 ) 1 3 : 27 4 = 27 × 27 × 27 × 27 = 531441 Then 531441 1 3 = 3 √ 531441 = 81. That is deϐinitely a calculator job. But we could write 27 4 3 as ( 27 1 3 ) 4 = ( 3 √ 27 ) 4 = 3 4 = 81, which can be done in your head! 4.3 The ‘natural number’ e This section looks like a digression, but it is very important for the logarithms section that follows. It starts by looking at a number called e , which is sometimes known as Euler’s constant. Sections 4.3.1 and 4.3.2 could be omitted at an initial reading but will repay study at a later date. 4.3.1 What is e ? We begin with an innocent-looking data exercise on indices. Look at Data Exercise 4.1. The expression a x b y cannot be simplified. We calculate a x and b y separately and multiply them together. Remember these cube roots: 3 √ 8 = 8 1 3 = 2 3 √ 27 = 27 1 3 = 3 √ 64 = 64 1 3 = 4 3 √ 125 = 125 1 3 = 5 QUICKFIRE 4.3 Express 4 −2 as a decimal. QUICKFIRE 4.4 Calculate 2 −4 × 3 2 and express the answer as a decimal. QUICKFIRE 4.5 Without using a calculator, express (a) 64 2 3 and (b) 64 3 2 as numbers. QUICKFIRE 4.6 Without using a calculator, express 4 − 3 2 as a number. QUICKFIRE 4.7 (a) Express 72 as a m b n , in which a and b are prime numbers. (b) Use the third law of indices to express (72) 4 in the same way. QUICKFIRE 4.8 Express 125 √ 5 as 5 P . QUICKFIRE 4.9 Express 27 − 4 3 more simply without using a calculator. Mathematics for Physics 36