Passy's World of Mathematics

Expanding Two Brackets Binomials

Posted on June 24, 2013 by Passy

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Here at Passy’s World we use the “Crab Claws” method to Expand expressions which consist of two brackets.

It takes all the pain out of expanding these items, although messing with a real crab’s claws could cause some severe discomfort.

Why We Learn Expanding Brackets

Expanding Brackets does not achieve much on its own, but is a basic skill we need for doing a lot of other mathematical work.

Here at Passy’s World we developed an Algebra Equation for the Sydney Harbour Bridge in Australia.

We did this from scaling a photo we took of the Bridge, starting off by knowing that the bridge is 503 meters long.

This scaling involved using Ratio and Proportion. We took the number of pixels on the Photo which Adobe Photoshop gave us for the length of the Bridge, and divided this by 503m.

This gave us a Scale Factor value of 0.81 Pixels per Meter.

We could then apply this multiplying scale factor to all of our other Pixel lengths, and thereby convert them into meters.

To produce the graph of the Bridge, we needed to expand our brackets equation into a simplified equation that could then be entered into an Online Graphing Application.

We could then prove that our equation for the Bridge was correct, because its graph exactly matched the real life shape of the Bridge.

Accurate Graphing is just one reason why we need to know how to expand brackets. We also need to know how to expand brackets to make it easier to solve Algebra Equations.

If you would like to know more about the mathematics of the amazing Sydney Harbor Bridge, then check out our previous lesson all about this at the following link:

http://passyworldofmathematics.com/sydney-harbour-bridge-mathematics/

Expanding Two Brackets Containing Numbers

The easiest way to do these is using BODMAS of Pemdas, but there is another method known as “Binomial Expansion” which will also get us the correct answer.

Using the Order of Operations gives us the answer of 45.

Now let’s look at how we can get the same answer of 45, but by using the “Binomial Expansion” method.

This method involves multiplying and adding together the four numbers in our brackets in a specific set order.

We would not want to use the above method in real life, because it is much more work than using BODMAS or Pemdas.

HOWEVER: This longer “Binomial Expansion” method gives us a way of expanding brackets which contain Algebra letter variables.

We cannot use BODMAS or Pemdas on Algebra brackets, and they need to be done using the Binomial Expansion Method.

The “Crab Claws” Method

An easy way to remember the Binomial Expansion Pattern, is to use the “Crab Claw” method shown above.

We suggest you always draw out the two crab claws onto the brackets first, and then do all of the multiplying and adding of items.

Binomial Expansions Examples

Our first example involves expanding the expression: (h + 3) (y + 5)

Notice that we started off with four different items in the brackets: the h, 3, y, and 5.

Because we have done four multiplications during the Crab Claws expansion, we end up with a four part answer.

This final answer is simply left as a four term Algebra Expression, and it cannot be simplified any further.

Here are some further examples of Binomial Expansions using the Crab Claws method.

This example was a little trickier than our first one, because it has a minus sign subtraction in one of the brackets.

The four items in (k + 2)(v – 1) are k, 2, v, and -1.

This means that we get some Integer Multiplications to do in the expanding, and so we need to remember and apply our Integer Multiplication Rules:

These Integer Multiplication Rules can be summarised as follows:

The following example for Expanding (m-2)(n-6) contains the items: m, -2, n, and -6, which will mean we have some Integer Multiplications to do.

In this next example, we have some slightly larger terms to deal with, but we still use the same “Crab Claws” method to get our final answer.

Expanding and Answer Sizes

In Examples 1 to 4 that we have done so far, we had four different items in our brackets at the start, and this resulted in four separate terms in our final answer.

HOWEVER: Not all Binomial Expansions have a Four Part Answer.

For some Binomials we are able to add or subtract some “Like Terms”, and reduce down to a “Three Part” final answer, or even in some cases a “Two Part” final answer.

If you are not sure about how to do “Like Terms” simplifying, then review “Combining Like Terms” using our previous lesson at the link below:

http://passyworldofmathematics.com/combining-like-terms/

For Binomial Expansions, we can get four, three, or even two part answers; depending on how many “Like Terms” we had back in our starting brackets.

The following examples show how we get “Three Part” and “Two Part” final answers.

Like Terms Binomial Expansions

For the brackets: (m + 4) (m + 1) we only have three different items: the m, 4, and 1.

This results in a “Three Part” final answer as shown below.

The Binomial: (b – 3) (b – 2) is another example where we get a three part answer.

The original brackets only contain three different starting items: the b, -3 and -2.

Here is another example which produces a “Three Part” final answers.

Two Part Final Answer Binomial Expansions

Whenever we have (Item One + Item Two) (Item One – Item Two) we get a final answer which contains only two terms.

There is even a Shortcut Rule for this called the “Difference of Perfect Squares” or “D.O.P.S. Rule.

Here is a typical example of this type of expansion.

Larger Items Binomial Expressions

Sometimes our starting brackets contain not just single letter items like “y” or “a” or “m” etc, but instead we have larger items such as “3y” or “2a” or “-5m”.

This is not really an issue at all, we simply do our “Crab Claws” method expansion as usual, and simplify any like terms that are produced along the way.

The following example shows how to do one of these Larger Items Expansion

Expanding Squared Brackets

When our starting question has one set of brackets squared, we need to first write it out as two brackets.

Five Squared means 5 x 5 and so (y + 5)^2 simply means (y + 5)(y + 5) .

Once we make this starting adjustment, we can then do our “Binomial Expansion” as normal using the “Crab Claws” method.

The following is a typical example of this type.

Note that there is a shortcut rule for these type of expansions which is called the “Perfect Squares Rule”.

However, here at Passy’s World we find it much easier to just use the “Crab Claws” method, and so we have not covered this shortcut rule in this lesson.

“FOIL” Rule for Binomial Expansions

You may hear of the “Crab Claws” pattern being called the “F.O.I.L.” Rule.

FOIL stands for “First” “Outside” “Inside” “Last” and is just another way of describing the four step expansion pattern that we use.

If you would like to see the “FOIL” version of the “Crab Claws” method in action, then check out the following video.

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Posted in Algebra, Equations | Tagged Algebra, algebra expansion, binomial, binomial expansion, crab claw method for expanding, crab claws rule, crab claws rule expansion, distributive law, expand brackets, expanding binomials, expanding using crab claw rule, expanding using crab claws, expansion, FOIL, foil expansion, foil pattern for expanding, how to do binomials, how to expand binomials, how to expand brackets, how to expand two brackets, what is a binomial | 1 Comment

Transposing and Rearranging Formulas

Posted on May 28, 2013 by Passy

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In a previous lesson we showed how to Solve Equations using the work down through the “Onion Skins” Method.

Knowledge of that lesson is needed as a background, before doing our lesson on Rearranging Formulas.

Click here for Onion Skins Equations Lesson

In our Transposing Formulas lesson we will review the use of the “Onion Skin” method, and show how it can be applied to Transposing (or rearranging) Algebra Formula Equations.

Review of Onion Skin Method

Here are the steps that we follow to solve an Equation using Onion Skins.

Here is an example of solving a simple Equation using the Onion Method.

Onion Skins for Transposing

The steps we do for Transposing an Equation into a new equation with a different letter variable as the subject, is basically the same as we do for Solving Equations.

Transposing Equations is the same as Solving Equations, except that we have a set of letter variables to deal with, and very few if any numbers present.

Here is a simple Transposing Example that is very similar to our previous Solving Equation example.

Solving a Two Step Equation Using Onion Skins

The following example shows how to solve a two step equation.

We always put our first innermost circle around the letter variable we are solving for, in this case it was “h”.

In the above example, the variable letter “h” is x2 and +3.

If you are not sure which of these two operations gets circled first, remember we have to do them in BODMAS/PEMDAS order.

This means we circle the 2h for x2 before we circle the 2h + 3 for the +3 operation.

Transposing a Two Step Formula Using Onion Skins

Let’s now look at a Formula that is very similar to the Equation we just solved, but contains lots of letters instead of numbers.

Formulas Containing Exponents and Fractions

The following examples show to transpose formulas that contain exponents such as power of 2 squares.

Squaring comes under the “O” for other things in BODMAS, and under the “E” for Exponents in PEMDAS.

Because it is the next operation to follow brackets, it is usually one of the very first onion skins we draw around our letter.

Square Root is the opposite of Squaring, and so when we peel the Onion we do square root to undo squaring.

The next example contains a Square power of 2 exponent, but because it is not on the letter we are transposing to, we do not have to reverse it out with a square root.

The formula has a Fraction and we need to move it into the bottom of our Algebra Formula BEFORE we draw our Onion Skins.

In the next example, we are using the same formula, but this time we are transposing to make “v” the subject.

This “v” that we are transposing to does have a square on it, so we will have to do a square root reversal of it when we “peel” our onion.

(Squaring and Square Roots are opposites, just like + and – are opposites).

Transposing Financial Mathematics Formulas

Financial Mathematics has a lot of Cost Price, Selling Price, Markup, Discount, Profit and Loss Formulas.

Rather than memorising every single possible formula we may need, it is much easier to only know the main formulas, and then be able to transpose these to get any other formulas we need.

In the following example, we start with the Formula for applying a discount “D”, to a Marked Price on an item “M” to work out the selling price “S”.

We transpose the formula to an M= formula, so that we could work out the original Marked Price required on an item if it is going to sell for a certain amount when discounted.

Notice that we follow “BODMAS/PEMDAS” order, and onion circle the brackets item before circling the dividing part of the formula.

Transposing a Multi-Variable Formula

In this next example we have quite an involved formula, which results in many onion skins that have been drawn outwards from our subject letter of “c”, in BODMAS/PEMDAS order.

Transposing a Square Root Formula

The following formula contains a square root, and when we reverse this out in the peeling process we use squaring as its opposite.

The above transposing is probably the most complicated example we have covered.

Note that because the k/l is all under the Square Root sign it is circled before the actual Square Root sign.

This does not seem to follow BODMAS/PEMDAS, because the “D” dividing should come after the “O” other things square root.

The k/l being under the Square Root sign is kind of like it being in brackets, and it must be considered first, before a Square root can be taken.

Eg. Square Root of 27/3 would be worked out as Square Root of 27/3 = Square Root of 9 = 3; and not as Square Root of 27 then divided by 3.

Transposing When Subject is in Denominator

If our desired subject letter varibale is inthe bottom of a Fraction, we cannot directly apply the onion skins.

We must first flip over both sides of the Equation, and then do our Onion Skins.

This is mathematically Possible because things like 12/6 = 6/3 are still true when we flip both sides and have 6/12 = 3/6.

Related Items

Solving Equations Using Onion Skins

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Posted in Algebra, Equations, Exponents | Tagged Algebra, algebra formula, algebra formulae, algebra formulas, bodmas formulas, changing the subject in formulas, Equations, formulas, how to do rearranging for equations, how to reaarange an equation, how to rearrange a formula, how to transpose a formula, how to transpose an equations, making subject in formula, new subject in formula, onion method for algebra, onion method for equations, onion skin method for formulas, pemdas formulas, rearranging formulas, rearranging forumulae, transposing formulae, transposing formulas | Leave a comment

Significant Figures

Posted on May 25, 2013 by Passy

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Often our Calculator gives us very long answers which need to be rounded off to have a smaller number of “Significant Figures”.

For example, if we measure the Height of all the students in our class, and use a calculator to get an average, we might get an answer like 172.3421 cm.

If we record this long answer as our average, then it imples we measured the students’ heights to an accuracy of 4 decimal places.

There is no way we are allowed to claim this level of accuracy, (unless we used $20 000 worth of very high tech laser measuring equipment!).

We probably measured the heights with a ruler or a tape measure, and they are only accurate to the nearest whole number.

Our final answer for the Average Height neeeds to be written as 172cm, not 172.3421 .

Other real life situations also need us to think about how accurately a value has been measured.

The actual number of the people attending the above event might have been 14 687 or perhaps 15 215.

However we only need to know the figure to the nearest thousand, and so it is expressed as 15 000 or 15 thousand.

The 15 is the important part of the value, the three zeroes on the end just tell us that it is in units of thousands.

We say that 15 000 has two significant figures.

Significant Figures are used a lot in Science, Economics, Statistics, Finance, and many other areas of life where we are measuring things to a certain level of accuracy.

Significant Figures and Exact Values

The Significant Rules figures we are covering in this lesson must never be applied to “Exact Values”.

In the previous example of the estimated crowd size, we were given the five figure value of 15 000, and worked out that it had only two Significant figures.

This is the correct answer for 15 000 as a measured value in this particular situation.

However, the value of 15 000 does not always have two Significant Figures.

If we have an EXACT VALUE, like you are borrowing $15 000 from the bank for a Car, then this needs to be exactly $15 000, (not $14 821 or $15 100 or any other value).

A $15 000 transaction with the Bank is always a 5 Significant Figures value, because it involves EXACTLY $15 000 (and is not an estimated or rounded off value).

A Freeway Speed Limit of 100 km/hr is also an Exact Value, and has three significant figures.

An Exam Pass cutoff level of 50% is an exact value, and has two significant figures.

Significant Figures Rules

There is a set of fairly complicated rules for working out how many “Significant Figures” are in a given number or decimal value.

Here at Passy World we find it easier to use Scientific Notation to work out the number of Significant Figures, rather than use all of these rules.

However if you would like to use these rules, then check out the following video which shows you how.

The video at the link below also covers these five Significant Figures rules extremely well:

Click Here for Signicant Figures Video

Using Scientific Notation for Significant Figures

If we put our number or decimal value into Scientific Notation, it is very easy to work out the number of Significant Figures.

Once our number or decimal is in Scientific Notation, all we do is count how many digits there are!

If you do not know how to do Scientific Notation, then check out our lesson on this at the link below:

Click here for our lesson on Scientific Notation

Important Helpful Hints

Whenever doing Scientific Notation and Significant Figures, always keep the following in mind.

Normal Numbers bigger than 1, or large numbers,always have a POSITIVE Power of 10.

Values smaller than 1, usually decimal values,always have a NEGATIVE Power of 10.

The first part of Scientific Notation is always a number value that is between 1 and 10. (eg. 1, 2.345, 3.65, 6.310, 7.0, 8.5, 9.9999 etc)

The second part of Scientific Notation is a Power of 10 which tells us how many places the decimal point is moving.

The resulting number of digits in our 1 to 10 number is the number of Significant Figures.

Determining Significant Figures Examples

The following examples show how we can put a number or decimal value into Scientific Notation, and then very easily determine its “Number of Significant Figures”.

We can also determine the number of Significant Figures in Decimal Values.

The following example shows a very important exception rule for removing zeroes from Decimal values that we are converting into Scientific Notation.

ANY ZERO THAT CAME AFTER THE DECIMAL POINT IN THE ORIGINAL STARTING DECIMAL NUMBER MUST NOT BE REMOVED.

It must not be removed because it is a Significant digit.

The reason why is a little complicated, but works like this.

If somebody wrote down a measurement as 0.0050, this means they measured it to four decimal places to the nearest ten thousandth, and it is 50/10000. The “50” in 50 thousandths has two Significant Figures.

If they had only measured it to three decimal places to the nearest thousandth, then the value would have been recorded as 0.005 and it is 5/1000. The “5” in 5 thousandths has one Significant Figure.

Here is another example that has an ending decimal zero value which is Significant and must not be removed.

Significant Figures and Rounding Off

If we convert our number or decimal value into Scientific Notation, then we can round if off using normal rounding off rules.

Rounding Rules involve the following:

Work out where we are rounding off to, and then look at the next digit after this:

If this next digit is 0 to 4 (0, 1, 2, 3, 4 ) then “Do no More” and discard all the extra not needed digits.

If this next digit is 5 to 9 (5, 6, 7, 8, 9 ) then “Up this Time” and add on one, and then discard all the extra not needed digits.

Just remember: “0 to 4 Do No More”, or “4 and below, Let it Go”.

To review Rounding Off if you are not too sure about how it works, then check out our lesson on Rounding Off at the following link:

Click here for Rounding Off Lesson

Rounding Off Examples

The steps to Rounding off are:

Convert our given Value into Scientific Notation

Round off to the required Significant Figures

Convert the rounded off Scientific Notation back to a normal Number or Decimal values.

In the following two examples we are Rounding Off to TWO Significant Figures.

In the following two examples we are Rounding Off to THREE Significant Figures.

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Scientific Notation

Posted on May 24, 2013 by Passy

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“Scientific Notation”, (also referred to as “Standard Form”), is used by Scientists to represent very large numbers, like the Distance between stars and planets, in a much simpler form.

It is also used by Geographers and Economists to represent very large numbers such as how many square kilometers in area a Country is, and how big in millions its population is, or how many billions of dollars its current debt level is at.

Format of Scientific Notation

Scientific Notation always involves having a number that is between 1 and 10 multiplied by a Power of 10.

For example the distance from the earth to the sun is 149 600 000 km, and because this number is so big, it is usually listed in Scientific Notation.

Helpful Tips for Dealing with Scientific Notation

Here are a couple of very useful tips to constantly keep in mind when doing any activities that involve Scientific Notation.

Videos About Scientific Notation

The following five minute video is an excellent explanation about why people use Scientific Notation.

Here is a quick video on how to convert Scientific Notation into a standard number value.

The following vidoe shows how to convert number values into Scientific Notation

Here is a quick video about how to write decimal values in Scientific Notation

Scientific Notation into Numbers

The following examples show how to convert Scientific Notation expressions into the actual numbers or decimals that they represent.

Numbers into Scientific Notation

Converting Numbers into Scientific Notation is a little bit trickier than doing Scientific Notation into Numbers.

However we should be okay if:

We follow the set sequence of steps, (shown in the examples below) and

Remember our key facts from the “Helpful Tips”:

Normal Numbers bigger than 1, or large numbers, always have a POSITIVE Power of 10.

Values smaller than 1, usually decimal values, always have a NEGATIVE Power of 10.

The first part of Scientific Notation is always a number value that is between 1 and 10. (eg. 1, 1.5, 2.3, 3.167, 4, 5.21, 6.3, 8.20, 9.9999 etc)

The second part of Scientific Notation is a Power of 10 which tells us how many places the decimal point is moving.

Here are some examples showing the full working out steps.

Because it can be easy to make little mistakes when converting into Scientific Notation, it is always an excellent idea to check our answers by working backwards and checking that our Scientific Notation does give us the back the original number or decimal that we started with.

It doesn’t take long to do this check, it is easy to do, and it guarantees us that we have the correct answer.

Here is how we can do a simple check for the example we just did above.

Here is an example showing how to convert a Decimal Value into Scientific Notation.

REMEMBER that Decimal Values always produce Negative Powers of 10.

The following example shows a very important exception rule for removing zeroes from Decimal values that we are converting into Scientific Notation.

ANY ZERO THAT CAME AFTER THE DECIMAL POINT IN THE ORIGINAL STARTING DECIMAL NUMBER MUST NOT BE REMOVED.

The reason why is a little complicated, but works like this.

If somebody wrote down a measurement as 0.0050, this means they measured it to four decimal places to the nearest ten thousandth, and it is 50/10000.

If they had only measured it to three decimal places to the nearest thousandth, then the value would have been recorded as 0.005 and it is 5/1000.

If we have a Power of 10 like 10, 100, 1000, 10 000 etc, we can convert it into Scientific Notation as shown below.

If we have a value which is between 1 and 10, then we can express it in Scientific Notation using the Power of Zero.

This is shown in the example below.

Scientific Notation on Calculators

There are usually special buttons on calculators for entering values that are in Scientific Notation format into the calculator.

Related Items

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Zero and Negative Exponents

Posted on May 21, 2013 by Passy

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

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:

Click Here to Learn About Exponential Decay

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.

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.

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.

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.

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

Negative Exponent Rule

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.

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:

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.

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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Expanding Two Brackets Binomials

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Zero and Negative Exponents

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