Cambridge International Mathematics

Exponents and surds (Chapter 6) 137

Discovery Properties of surds

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Notice that

p

4 £9=

p

36 = 6 and

p

4 £

p

9=2£3=6, which suggests that

p

4 £

p

9=

p

4 £ 9 :

Also,

r

36

4

=

p

9=3 and

p

36

p

4

=

6

2

=3, which suggests that

p

36

p

4

=

r

36

4

.

What to do:

Test the following possible properties or rules for surds by substituting different values ofaandb. Use

your calculator to evaluate the results.

1

p

a£

p

b=

p

ab for all a> 0 ,b> 0.

2

qa

b

=

p

a

p

b

for all a> 0 ,b> 0.

3

p

a+b=

p

a+

p

b for all a> 0 ,b> 0.

4

p

a¡b=

p

a¡

p

b for all a> 0 ,b> 0.

You should have discovered the following properties of surds:

²

p

a£

p

b=

p

a£b for a> 0 , b> 0

²

p

a

p

b

=

r

a

b

for a> 0 , b> 0

However, in general it is not true that

p

a+b=

p

a+

p

b or that

p

a¡b=

p

a¡

p

b:

Example 18 Self Tutor

Write in simplest form:

a

p

3 £

p

2 b 2

p

5 £ 3

p

2

a

p

3 £

p

2

=

p

3 £ 2

=

p

6

b 2

p

5 £ 3

p

2

=2£ 3 £

p

5 £

p

2

=6£

p

5 £ 2

=6

p

10

F PROPERTIES OF SURDS [1.10]

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Y:\HAESE\IGCSE01\IG01_06\137IGCSE01_06.CDR Monday, 15 September 2008 3:02:50 PM PETER