Having discussed nth roots, we are now in a position to define the expression ax, where a is positive and x is a rational power (or exponent).

## Definition

If aÂ >Â 0, m Author(s): The Open University

This unit explores reasons for studying mathematics, practical applications of mathematical ideas and aims to help you to recognize mathematics when you come across it. It introduces the you to the graphics calculator, and takes you through a series of exercises from the Calculator Book, Tapping into Mathematics With the TI-83 Graphics Calculator. The unit ends by asking you to reflect on the process of studying mathematics.

In order to complete this unit you will need
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Learning outcomes

By the end of this unit you should be able to:

• Section 1: Sets

• use set notation;

• determine whether two given sets are equal and whether one given set is a subset of another;

• find the union, intersection and difference of two given sets.

• Section 2: Functions

• determine the image of a given function;

• determine whether a given function is one-one
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Acknowledgements

All written material contained within this unit originated at the Open University.

Except for third party materials and otherwise stated (see terms and conditions), this content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence

1. Join the 200,000 students currently studyi
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3.2.1 Try some yourself

1 Use the method outlined in Example 9 to estimate each of the following, and then use yo
Author(s): The Open University

2.2.1 Try some yourself

1 Find the following powers by hand, as estimates for calculator work.

• (a) 107

• (b) 108

• (c) 34

• (d) (âˆ’2)2
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1.3 Square roots

Given any number, you now know how to find its square. But, given the squared number, how do you find the original number?

## Example 3

Learning outcomes

By the end of this unit you should be able to:

• evaluate the squares, cubes and other powers of positive and negative numbers with or without your calculator;

• estimate square roots and calculate them using your calculator;

• describe the power notation for expressing numbers;

• use your calculator to find powers of numbers;

• multiply and divide powers of the same number;

• understand and apply negative powers, t
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1 Modelling static problems

The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook. Once you have completed the workbook and exercises return to this page and watch the video below, â€˜The arch never sleepsâ€™, which discusses a practical application of some of the ideas in workbook.

Click 'View document' to open the workbook (PDF, 0.8
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1.1 What is 'globalisation'?

### Activity 1 What does â€˜globalisationâ€™ mean to you?

Note down on paper or in your learning journalÂ  your first tho
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3.2 The past temperature of the planet

Measuring the concentration of lead in the ice is called a direct measurement: the ice sample is melted and the water produced contains a very small but readily measured quantity of lead dust. A very accurate set of scales would be needed to measure it, but it is a directly measured quantity. There are also many indirect measurements that can be made using proxy data. The concept for using proxies is both simple and brilliant: one measured property allows inference about other states o
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1 A climate change icon

The polar bear has become an international climate change icon. But how much is known about this bear, its habitat and life? This unit will talk about the role of language, but by way of introduction how about the name of this bear? To me it is the polar bear; to a German it is an EisbÃ¤r (ice bear) and to a French person it is an ours blanc (white bear). In these three examples the bear is referred to as polar, white, or an ice bear â€“ eminently sensible. The Latin name for th
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Acknowledgements

Grateful acknowledgement is made to the following sources for permission to reproduce material in this unit:

The content acknowledged below is Proprietary and used under licence (not subject to Creative Commons licence). See Terms and Conditions.

## Unit image

Getty photo disc

2. Enjoyed this? Browse through our host of free module materials on LearningSpace<
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2.2.2 Environmental economics and green consumerism

In economic terms, green consumerism is typically expressed using measures based on the willingness to pay (WTP) principle. As mentioned above, this takes two main forms: eco-taxation, in which environmental costs are estimated and added to the price of commodities (e.g. vehicles with high carbon emissions); and eco-labelling, in which products are labelled with relevant environmental information, such as is now required by the food industry and governments in many industrialised count
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5 Human influence?

Evidence suggests that global temperature is beginning to rise. There are several factors that could cause this. Only one is affected by human activity.

Click on 'View document' to view a chart showing the rates of energy gain and loss by the Earth's surface and atmosphere

6 Population growth

Earlier it was stated that three factors check population growth. These are predation, disease and insufficient food supply. For much of our history, our ancestorsâ€™ numbers were indeed limited by wars, disease and famine. The world population remained relatively stable until around 300 years ago. Then at the beginning of the 19th century (100 years after population growth started its geometric increase), the demographer Thomas Malthus predicted that population growth would outstrip food pro
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5.1 Pollution and loss of biodiversity

Unfortunately, halting the disappearance of species cannot be achieved simply by measures such as putting fences around special habitats and asking people not to pick the flowers or disturb the breeding birds. Many species are vanishing because of pollution. You probably have a good understanding of the meaning of this term, but it is variously defined. The Open University has a course on environmental control and public health, in which pollution is defined as the introduction into th
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4.5 Global climate change

I would like to turn now to the possible consequences of our use of energy for global climate change. Our pattern of energy use relies heavily on burning carbon-based fossil fuels, releasing carbon dioxide which spreads evenly around the globe and builds up slowly in the atmosphere. Carbon dioxide is a greenhouse gas, which means that it has the potential both to warm the atmosphere and to change our global climate. It is not the only greenhouse gas but is the most important of those e
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4.3 The UK experience: competing trends

But one striking example does not make an argument. To try to get a fuller and possibly fairer picture of energy use by domestic refrigerators I'd like also to look at the UK experience over the past few decades.

To start with it helps to have a feel for which parts of the UK economy use the most energy. The UK Department of Trade and Industry (DTI, 1998), identifies four main economic sectors: domestic (households), industry, services and transport.

In 2003 the domestic sector (h
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2.1 Environment and technology

A central concern of environmental studies is the relationship between technology and our environment: how people use technology to transform materials into forms which can meet our needs and wants. In the process of doing this we inevitably change the environment which provides these materials but which also supports all life.

A few moments ago I went to my fridge and took some milk out to add to a cup of coffee. I used this common example of a modern domestic appliance without a secon
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