1. Concepts of Computation
Session 7a
Functions
Dr Oded Lachish
(Slides prepared with the support of Dr Paul Newman and Eva Szatmari)

2. Functions
Exponential functions laws and simplification

3. Exponential function simplification Laws of indices

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

5. 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!)

6. 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

7. Laws of indices – more multiplication examples
4) ax * ay = a(x + y)
42 * 43 = 4(2+3) = 45
16 * 64 =
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

8. 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 7−5 = 10 2
As can easily be checked by doing it longhand!

9. 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

10. Power of Zero Take an number as an example, 108
If you divide any number by itself you get 1
Using our rules:
= 10 8−8 = 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

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

12. 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

13. 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
2-2 = = 1 4
2-3 = = 1 8 …
2-n = 𝑛 = 1 2∗2∗…with 𝑛 2𝑠
Exponents and Logarithms are opposites of each other

14. 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 =
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

15. 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

16. 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* =
X times
X times
X times
y times

17. 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

18. Roots and Fractional Powers II
Any number to the power 1 is itself, so, taking 10 as an example
10 1 = = *
= * 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 = 3 10
And so on

19. Working with fractional powers
Use fractional powers as you would any other number, for example
10 − = ( 10 − ) 3 = (10 3 ) − = ( 10 −3 ) ; they all mean the same thing
Note that you can write the fractional power in decimal form, just as = 0.5
= and 10 − = 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
= 2
16 − = 16 −1.5 = ( 16 −0.5 ) 3 = ( 1 4 ) 3 = = 1 64

20. 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