Exponents and surds (Chapter 6) 125Historical note
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1=1^3
3+5=8=2^3
7+9+11=27=3^3 etc.Nicomachus of Gerasa lived around 100 AD. He discovered
an interesting number pattern involving cubes and sums of odd
numbers:NEGATIVE BASES
So far we have only consideredpositivebases raised to a power.We will now briefly look atnegativebases. Consider the statements below:(¡1)^1 =¡ 1
(¡1)^2 =¡ 1 £¡1=1
(¡1)^3 =¡ 1 £¡ 1 £¡1=¡ 1
(¡1)^4 =¡ 1 £¡ 1 £¡ 1 £¡1=1(¡2)^1 =¡ 2
(¡2)^2 =¡ 2 £¡2=4
(¡2)^3 =¡ 2 £¡ 2 £¡2=¡ 8
(¡2)^4 =¡ 2 £¡ 2 £¡ 2 £¡2=16
From the pattern above it can be seen that:² anegativebase raised to anoddpower isnegative
² anegativebase raised to anevenpower ispositive.Example 3 Self Tutor
Evaluate:
a (¡2)^4 b ¡ 24 c (¡2)^5 d ¡(¡2)^5a (¡2)^4
=16b ¡ 24
=¡ 1 £ 24
=¡ 16c (¡2)^5
=¡ 32d ¡(¡2)^5
=¡ 1 £(¡2)^5
=¡ 1 £¡ 32
=32CALCULATOR USE
Power keysx^2 squares the number in the display.^ raises the number in the display to whatever
power is required. On some calculators this
key is yx , ax or xy.Notice the effect of
the brackets in
these examples.Not all calculators will
use these key sequences.
If you have problems,
refer to the calculator
instructions on page 12.Different calculators have different keys for entering powers, but in
general they perform raising to powers in a similar manner.IGCSE01
cyan magenta yellow black(^05255075950525507595)
100 100
(^05255075950525507595)
100 100
Y:\HAESE\IGCSE01\IG01_06\125IGCSE01_06.CDR Monday, 15 September 2008 2:30:09 PM PETER