Cambridge International Mathematics

Exponents and surds (Chapter 6) 135

Historical note

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The namesurdand theradicalsignp both had a rather absurd past. Many

centuries after Pythagoras, when the Golden Age of the Greeks was past,

the writings of the Greeks were preserved, translated, and extended by Arab

mathematicians.

The Arabs thought of a square number as growing out of its roots. The roots

had to be extracted. The Latin word for “root” isradix, from which we get

the wordsradicaland radish! The printed symbol for radix was firstR, thenr,

which was copied by hand as p.

The wordsurdactually came about because of an error of translation by the

Arab mathematician Al-Khwarizmi in the 9 th century AD. The Greek word

a-logosmeans “irrational” but also means “deaf”. So, the Greeka-logoswas

interpreted as “deaf” which in Latin issurdus. Hence to this day,irrational

radicalslike

p

2 are calledsurds.

BASIC OPERATIONS WITH SURDS

We have seen square roots and cube roots in previous courses. We can use their properties to help with

some simplifications.

Example 15 Self Tutor

Simplify:

a (

p

5)^2 b

μ

1

p

5

¶ 2

a (

p

5)^2

=

p

5 £

p

5

=5

b

μ

1

p

5

¶ 2

=

1

p

5

£

1

p

5

=^15

Example 16 Self Tutor

Simplify:

a (2

p

5)^3 b ¡ 2

p

5 £ 3

p

5

a (2

p

5)^3

=2

p

5 £ 2

p

5 £ 2

p

5

=2£ 2 £ 2 £

p

5 £

p

5 £

p

5

=8£ 5 £

p

5

=40

p

5

b ¡ 2

p

5 £ 3

p

5

=¡ 2 £ 3 £

p

5 £

p

5

=¡ 6 £ 5

=¡ 30

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