Concepts of Computation Session 7a Functions Dr Oded Lachish Email: oded@dcs.bbk.ac.uk (Slides prepared with the support of Dr Paul Newman and Eva Szatmari)

Functions Exponential functions laws and simplification

Exponential function simplification Laws of indices

Manipulating exponentials Suppose we have to multiply to numbers in exponential form Look for the pattern: 22 * 26 = 4 * 64 = 256 = 28 And 2 + 6 = 8 103 * 104 = 1000 * 10000 = 10000000 = 107 And 3 + 4 = 7

Seeing the pattern: multiplication The rule seems to be: To multiply together two numbers in exponential form Add the indices For example 28 * 27 = 28+7 = 215 (check it yourself!)

Rule for multiplying numbers in exponential form For any two numbers, to the same base a, the law of multiplication is: ax · ay = a(x + y) So, 33 * 34 = 33+4 = 37 28 * 23 = 28+3 =211 32 * 54? Can’t apply the rule! Why not? Different bases, you can apply the rule unless the numbers are to the same base

Laws of indices – more multiplication examples 4) ax * ay = a(x + y) 42 * 43 = 4(2+3) = 45 16 * 64 = 1024 32 *33 = 3(3 + 2) = 35 9 * 27 = 243 52 * 5-1 = 5(2-1) = 51 25 * (1/5) = 5 Rule 1. Anything raised to the power of 0 is 1

Rule for dividing numbers in exponential form The rule is similar: To divide two numbers, expressed in exponential form to the same base, subtract the indices For example 10 7 10 5 = 10 7−5 = 10 2 As can easily be checked by doing it longhand!

Laws of indices – division examples ax / ay = a(x - y) 105 / 103 = 10(5-3) = 102 100,000 / 1,000 = 100 27 / 23 = 2(7-3) = 24 128 / 8 = 16 64 / 4 = 26 / 22 = 2(6- 2) = 24 = 16 Rule 1. Anything raised to the power of 0 is 1

Power of Zero Take an number as an example, 108 If you divide any number by itself you get 1 Using our rules: 10 8 10 8 = 10 8−8 = 10 0 =1 This will work for any number, so Any number, raised to the power of 0 equals 1 For all bases a, a0 = 1

Examples 1500 = 1 (-298)0 = 1 (1 / 3)0 = 1 Π0 = 1

Negative Powers Take an example: 103 Clearly, 10 3 ∗ 1 10 3 = 1 i.e. any number multiplied by its inverse gives 1 What do we mean by 10-3 ? Apply the rule just learned 10 3 ∗10 −3 =10 3−3 =10 0 =1 i.e. A negative power indicates the inverse of a number

Exponents showing negative powers 20 = 1 21 = 2 22 = 2 * 2 = 4 23 = 2 * 2 * 2 = 8, … 2n = 2 * 2 * … with n 2s 2-1 = 1 2 1 = 1 2 2-2 = 1 2 2 = 1 4 2-3 = 1 2 3 = 1 8 … 2-n = 1 2 𝑛 = 1 2∗2∗…with 𝑛 2𝑠 Exponents and Logarithms are opposites of each other

Negative Indices for powers of 10 For powers of 10, the negative index can also be represented in decimal form 3-1 = 2 zeros 10-3 = 0.001 10-1 = 9 zeros 10-10 = 0.0000000001 For a positive exponent, the number of zeros to the left of the decimal point = exponent For a negative exponent, the number of zeros to the right of the decimal place = exponent - 1

Power of a power To evaluate a number, already expressed as a power, to another power, simply apply the rules (102)3 = 102 * 102 * 102 = 10(2+2+2) = 10(2x3) = 106 To evaluate a number, already expressed as a power, to another power, multiply the powers together

Power of a power – general case (ax)y = axy (a multiplied by itself x times) multiplied by itself y times) = a multiplied by itself x ·y times (a *a *…) · ….(a *a *…) ………..…(a *a *…) Example: (22)5 = 22*5 = 210 = 1024 4*4*4*4*4 = 1024 X times X times X times y times

Roots and Fractional Powers I For any number 𝑎, if we can find another number which, when multiplied by itself gives the original number, we call it the Square Root of 𝑎 , written 2 𝑎 Similarly, a number which when multiplied by itself 3 times gives 𝑎, we call this the Cube Root of 𝑎, written 3 𝑎 So, 2 * 2 = 4 so 2 4 =2 Similarly 3 * 3 * 3 = 27 so 3 27 =3

Roots and Fractional Powers II Any number to the power 1 is itself, so, taking 10 as an example 10 1 = 10 1 2 + 1 2 = 10 1 2 * 10 1 2 10 = 2 10 * 2 10 We can identify the ½th power of a number with its Square Root Similarly a 1/3rd power is the Cube Root, or 10 1 3 = 3 10 And so on

Working with fractional powers Use fractional powers as you would any other number, for example 10 − 3 2 = ( 10 − 1 2 ) 3 = (10 3 ) − 1 2 = ( 10 −3 ) 1 2 ; they all mean the same thing Note that you can write the fractional power in decimal form, just as 1 2 = 0.5 10 1 2 = 10 0.5 and 10 − 3 2 = 10 −1.5 Of course, you can use powers of any base, we just used powers of 10 exclusively for convenience, here are a few examples to other bases 4 1 2 = 2 16 − 3 2 = 16 −1.5 = ( 16 −0.5 ) 3 = ( 1 4 ) 3 = 1 4 3 = 1 64

Summary – Laws of Indices a1 = a a0 = 1 a-x = 1/ax ax * ay = a(x + y) ax / ay = a(x – y) (ax)y = axy ax/y = 𝑦 𝑎 𝑥