# Zero and Negative Exponents

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In the above screenshot, we are using Adobe Photoshop to shrink down a large digital camera photo to one quarter of its original size.

We need to do this so that the photo can fit and load faster onto Facebook, or any other website we put it on.

Note that we can put full size digital camera photos onto the Web, but one of two things usually happens:

The App we upload the program with does the shrinking for us, but a lot of times not as well as we could do it ourselves in Photoshop

or

The App does not shrink the photo and the very large photo takes a long time to download and be resized by our web browser, and thereby makes the web page it is on load a lot slower.

So “Do the Math”, and always try to “optimise” the size and quality of any images, before you put them up onto the web.

Did you know that a computer cannot actually do Fractions or Percents during its processing?

To figure out what one quarter equals, at a low level the computer actually uses negative powers of 2, (associated with the “Binary” processing that all computerised devices use).

If it wasn’t for Negative Exponents, we would not have much of the wonderful photo processing that is currently available.

Negative Exponents which result in Power Fractions are also associated with “Exponential Decay”.

If you would like to find out more about Exponential Decay, then check out our previous lesson about “Exponents in the Real World” at the link below:

Today’s Exponents lesson is all about “Negative Exponents”, ( which are basically Fraction Powers), as well as the special “Power of Zero” Exponent.

Power of Zero Exponent

We can work out the number value for the Power of Zero exponent, by working out a simple exponent Division the “Long Way”, and the “Subtract Powers Rule” way.

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It can be seen from the above calculations that 2 to the power of zero equals 1.

We can also work out the numerical value of the Index Power of Zero, using patterns of Powers, as shown in the following diagram.

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From the work we have just done here, we can be quite certain that 2 to the power of zero equals 1. In fact ANY number or letter variable to the power of zero always equals 1.

Power of Zero Examples

No matter whether the value is a fraction, a really big number, or a variable letter, the answer is always “1” when this value is raised to the Power of Zero.

However, we need to be careful about exactly which part of a multi-item expression the Power of Zero actually applies to.

This is shown in the examples below.

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Negative Exponents

Negative Exponents are associated with Fractions.

We can prove this by working out a simple exponent Division the “Long Way”, and the “Subtract Powers Rule” way.

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Like we did with the Power of Zero Exponent, we can also do the Negative Exponent proof using Exponent Patterns.

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Negative Exponent Rule

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When we have a negative exponent on an item, we move it down to the bottom of the fraction, where it it then becomes a positive exponent power.

Negative Exponents – Examples

Shown below are some simple, and then some more complicated examples of how we simplify Negative Exponents by creating Fractions that have all of their exponent powers positive.

An expression is not fully simplified until all of its Negative Powers have been converted into Fraction Positive Powers.

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The last two examples in the above table are a little more involved than the 1/ fraction types.

We need to look at the “full story” on negative exponents and reciprocals.

The “Reciprocal” of a Number is the 1/ fraction of that number.

The “Reciprocal” of a Letter variable is the 1/ fraction of that letter variable.

The “Reciprocal” of a Fraction is the fraction flipped over so it is upside down – eg. 2/3 becomes 3/2.

Negative Exponents – “The Full Story”

The complete approach to dealing with all Negative Exponents is as follows:

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The Power of Power Rule, (as well as the Expanding Products and Quotients Rules), can also be used when we have Negative Exponents present.

Just remember: An expression is not fully simplified until all of its Negative Powers have been converted into Fraction Positive Powers.

Eg. There should never be any Negative Exponents left hanging around in our Answers.

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Videos About Zero and Negative Exponents

The following is a good Introductory Video on the basics of Negative Exponents.

This next video revises dividing exponents and then moves on to creating zero exponent and negative exponents.

It provides very good explanations, and shows plenty of examples, so is well worth watching.

The following Khan Academy video explains how zero and negative exponents arise from the patterns that are formed by powers.

Negative Exponents as “Opposites”

Another approach, which leads to the exact same situations we have covered up to now, is to use the idea of OPPOSITES.

With Positive Exponents we multiply the Base out as many times as the power number says to.

Negative is the OPPOSITE of Positive.

Division is the opposite of Multiplication.

This leads to the following Negative Exponents Rule:

FOR NEGATIVE EXPONENTS WE DIVIDE THE BASE AS MANY TIMES AS THE POWER NUMBER TELLS US TO.

Some people find this an easier way to deal with Negative Exponents.

The following thirteen minute video shows how Negative (Divide) Exponents are the Opposite of Positive (Multiply) Exponents.

It has a very detailed explanation, followed by several examples.

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