Cambridge Additional Mathematics

102 Surds, indices, and exponentials (Chapter 4)

Opening problem #endboxedheading

The interior of a freezer has temperature¡ 10 ±C. When a packet of peas is placed in the freezer, its

temperature aftertminutes is given by T(t)=¡10 + 32£ 2 ¡^0 :^2 t±C.

Things to think about:

a What was the temperature of the packet of peas:

i when it was first placed in the freezer ii after 5 minutes

iii after 10 minutes iv after 15 minutes?

b What does the graph of temperature over time look like?

c According to this model, will the temperature of the packet of peas ever reach¡ 10 ±C? Explain

your answer.

We often deal with numbers that are repeatedly multiplied together. Mathematicians useindices, also called

powersorexponents, to construct such expressions.

Indices have many applications in the areas of finance, engineering, physics, electronics, biology, and

computer science.

Aradicalis any number which is written with theradical signp.

Asurdis a real, irrational radical such as

p

2 ,

p

3 ,

p

5 ,or

p

6. Surds are present in solutions to some

quadratic equations.

p

4 is a radical, but is not a surd as it simplifies to 2.

p

ais the non-negative number such that

p

a£

p

a=a.

Important properties of surds are:

²

p

a is never negative, so

p

a> 0.

²

p

a is only real if a> 0.

²

p

ab=

p

a£

p

b for a> 0 and b> 0.

²

r

a

b

=

p

a

p

b

for a> 0 and b> 0.

Example 1 Self Tutor

Write as a single surd: a

p

2 £

p

3 b

p

18

p

6

a

p

2 £

p

3

=

p

2 £ 3

=

p

6

b

p

18

p

6

=

q

18

6

=

p

3

A SURDS

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_04\102CamAdd_04.cdr Tuesday, 14 January 2014 2:27:54 PM BRIAN