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When working with exponents, there are two new words that come up: exponent and base. As an example, in (x^2), 2 is the exponent while (x)is the base and in (4^{12}), 12 is the exponent while 4 is the base.

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Free worksheet with answer keys on exponents. Each one has model problems worked out step by step, practice problems, challenge proglems and youtube videos that explain each topic. J2SE 1.4.2 is in its Java Technology End of Life (EOL) transition period.The EOL transition period began Dec, 11 2006 and will complete October 30th, 2008, when J2SE 1.4.2 will have reached its End of Service Life (EOSL). Definition – What is the “4th Power” of a number? The “4th Power” of a number is the number multiplied by itself four times. Write it with a raised number 4 (the exponent) next to the base number. “number 4 “or “5 4 ” or “8 4 ” are examples of using an exponent 4.

Whole number exponents are about multiplication

As long as the exponent is a positive whole number, you can think of it as telling you how many times you should multiply the base by itself.

(2^3=2cdot2cdot2=8)
(x^3=xcdot xcdot x)
(5^2=5cdot5=25)

As you can see, (4^{12}) means “4 multiplied by itself 12 times” and that is a really big number. Many values that you calculate from exponents will be quite large.

There are a couple of things to note:

  • Anything with an exponent of zero is defined to be 1. So, (4^{0}=1), (100^{0}=1).
  • Ok right above, when I said anything, I meant “almost anything”. (0^0) is an indeterminate form. For everything else however, the rule above holds.
  • An exponent of 1 is the same as just writing the number by itself: (4^{1}=4).

Negative Exponents

Negative exponents are treated a bit differently. By definition, if we have a positive number (n) and a nonzero (b), then the following rule holds:

(b^{-n} = dfrac{1}{b^n})

Using this rule, if you see a value with a negative exponent, then it can be rewritten as 1 “over” that term with a positive exponent. Here are a few examples of how this works.

(3^{-2} = dfrac{1}{3^2} = dfrac{1}{9})

(10^{-1} = dfrac{1}{10^1} = dfrac{1}{10})

((-2)^{-4} = dfrac{1}{(-2)^4} = dfrac{1}{16})

As you can see, it is simply about following the formula when dealing with negative exponents. After this, you can simply calculate the value using the rules above.

Fractions as exponents

In algebra and more advanced math, it is very common to see an exponent of one-half or some other fraction. This is one way to represent terms involving roots, or radical terms. By definition (if n is nonzero):

(b^{frac{m}{n}}= sqrt[n]{b^m})

Working with these types of terms (radical terms) is a more advanced topic and will be the subject of a different article, but it is important that you be able to rewrite such terms using the definition as below.

(3^{frac{1}{2}}=sqrt{3^1}=sqrt{3})

(4^{frac{2}{3}}=sqrt[3]{4^2} = sqrt[3]{16})

As you study terms involving exponents, you will find that you need to combine rules quite often. For example, the rule above can be combined with the rule for negative exponents to simplify even more complex expressions.

Using a Calculator

Finding the value of terms with exponents on scientific or graphing calculators is quite easy. The usual key you will need will use a carat symbol ^ to represent that the next number is an exponent. So, (3^4) can be represented as 3^4 in most calculators. On the TI83 or TI84:

You can also always use the carat symbol on google (just type it into google! they now do automatic calculations for you) or wolfram alpha.

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Additional Reading

As you continue studying exponents, you will also want to review the order of operations. This lesson will show you how to handle expressions that have exponents, multiplication, and other operations all together.

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Installation (any Windows computers)

  • Download WinBUGS. Unzip this file into the place where you want to install it. This can be anywhere, for example, your Program Files directory. This should work for any version of Windows, for 32 or 64 bit. It includes the key for unrestricted use, and has been updated to version 1.4.3.

Easier installation (32 bit Windows only)

  • An easier way to install WinBUGS that only works on 32 bit machines is to download and install WinBUGS14.exe.
  • Then download and install the patch for 1.4.3.
  • Get the free key for unrestricted use by clicking here – this can be used for multiple installations.

Support and contact

See the Support and contact page for help resources.

Contents

  • Installing WinBUGS 1.4 in Windows

Introduction to WinBUGS

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WinBUGS is part of the BUGS project, which aims to make practical MCMC methods available to applied statisticians. See the main BUGS page for a summary of the different versions of BUGS available.

WinBUGS can use either a standard `point-and-click’ windows interface for controlling the analysis, or can construct the model using a graphical interface called DoodleBUGS. WinBUGS is a stand-alone program, although it can be called from other software. For a version that BUGS (BRugs) that sits within the R statistical package, see the OpenBUGS site.

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In the past, we have required that users register with us to obtain unrestricted use, in particular to give us some idea of the intended use of the program. This information has provided valuable feedback to ourselves and our funders. However, we no longer require users to register and the key for unrestricted use (of WinBUGS 1.4.x) can be downloaded from here.

Watch WinBUGS – The Movie! for a short Flash illustration of the basic steps of running WinBUGS. (This movie features WinBUGS 1.3 – when you’ve watched it you will understand why WinBUGS 1.4 includes scripts to avoid all the clicking and pointing!).

All the documentation specific to WinBUGS is available on-line and is packaged with the program.

Pdf versions of both the WinBUGS 1.4 manual (which DOES NOT include features in the 1.4.3 patch) and the GeoBUGS manual (which DOES feature the 1.4.3 patch) can now be downloaded, as can PDF versions of Volume 1, Volume 2, and Volume 3 of the examples.

Obtaining the File(s)

Please read the LICENCE AGREEMENT for WinBUGS, before downloading anything.

The files you need are WinBUGS14.exe (selecting the link will yield a dialog box inviting you to save the file onto your system) and the key for unrestricted use, which is free.

WinBUGS 1.1 and WinBUGS 1.2 are still available for those with historical interests. WinBUGS 1.3 and its key are also available and may still be useful if problems are obtained with WinBUGS 1.4 and for running PKBUGS Version 1 (see below).

Working

Installing WinBUGS 1.4 in Windows

For installation, run the file WinBUGS14.exe. One way to do this is as follows:

  1. Exit all other programs currently running (particularly if using Windows XP)
  2. Copy WinBUGS14.exe to your computer
  3. Go into Explore and double click on WinBUGS14.exe
  4. Follow the instructions in the dialog box
  5. You should have a new directory called WinBUGS14 within Program Files
  6. Inside the WinBUGS14 directory is a program called WinBUGS14.exe
  7. Right-click on the pretty WinBUGS icon, select `create shortcut’, then drag this shortcut to the desktop.
  8. Double click on WinBUGS14.exe to run WinBUGS14.

If you have problems after installation, check again that you did not have programs running. You could also try disabling any virus checker.

Note: There appears to be a problem with installing WinBUGS and/or various patches in Windows Vista. Vista doesn’t seem to like anyone overwriting files in the “C:Program Files” directory (regardless of permissions). Hence we recommend that WinBUGS be installed elsewhere, e.g. “C:”.

If all else fails (for example with a 64-bit machine), you can download a zipped version of the whole file structure and unzip it into Program Files or wherever you want it. WinBUGS makes no changes to the Registry.

Obtaining the key for unrestricted use

WinBUGS is free, but has absorbed a lot of time and grant money over the last two decades. We have tried to keep our employers happy with our time spent on the project by keeping track of how many people are downloading it and where they come from. It has also been very helpful to have an idea of what it is being used for. This information has, in the past, been collected via a registration process whereby users enter their details and a key for unrestricted use of the software has been sent to them via email (free of charge). However, as development shifts more and more towards OpenBUGS, we have decided that it is time to do away with this registration process and simply allow direct access to the key (which will no longer expire at the end of each year)…

Please click here to download the key for unrestricted use. After following the instructions given in the key, check that the Keys.ocf file in ..WinBUGS14BugsCode has been updated. (Some people have found they need to re-boot the machine to complete installation of the key.)

Upgrade to version 1.4.3

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A free patch for upgrading to version 1.4.3 is now available. Click here to obtain a full description and the patch file.

Some problems reported in WinBUGS 1.4

Some models that run in 1.3 are running slowly or not at all. Please let us have examples. This may be due to the change in the update order to graphical order, which has sorted out problems with forward sampling, but has introduced some new difficulties. We shall probably need to make adjustments to the adaptive slice sampler. (If your Trap message complains of problems with DFreeARS (Distribution-free adaptive rejection sampling), you could try changing to slice sampling for log-concave distributions – see manual for instructions.)

Etienne Rivot has pointed out that after a Trap, you can ‘Save States’ (model Menu), then reinitialise the model with these states. There may then, with luck, be a helpful proper error message!

Additions/corrections to WinBUGS 1.4 manual

  • Model specification/Formatting of data: When importing matrix data from Splus version 6 or above using the `dput’ command, you will need to replace Splus’s `nrow’ and `ncol’ by a `.Dim’ statement. The ordering of the dimensions remains as in the manual. The procedure for arrays remains as in the manual, except that Splus’s inverted commas and `.Dimnames’ statement should be removed.
  • An `inprod2′ function has been added in 1.4.3 – use this instead of `inprod’ to speed up runs.

Additions/corrections to GeoBUGS 1.2 manual

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  • The syntax given in the GeoBUGS manual for defining the scale of a map is wrong: In the manual we say that the syntax is:
    xScale: 1000
    yScale: 1000
    whereas, in fact, it should be
    Xscale: 1000
    Yscale: 1000
1.4 Working With Exponentsmr. Mac

What else is available for WinBUGS 1.4.3?

  • GeoBUGS 1.2 for spatial analysis
  • PKBUGS for pharmacokinetic modelling
  • A new test version for Reversible Jump MCMC.
  • Suites of S-plus/R functions for convergence diagnostics, suitable for use with any version of BUGS.
  • WinBUGS can be remotely called from a variety of packages (although note that one can now run BUGS from within R using BRugs).
  • We now have a website exclusively for development and distribution of extensions to WinBUGS – see the WinBUGS development site and Dave Lunn’s ISBA Bulletin article for more details.

From a technical perspective, WinBUGS uses the software philosophy of component-oriented progamming, which means the program is constructed from a set of cooperating components. This set is not closed, and so the functionality of WinBUGS can be continuously extended by developing new components: see the OpenBUGS site for the latest developments.

Any queries please contact bugs at mrc-bsu.cam.ac.uk