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= = = = = = =**__Exponents:__**=

This basically is a shorthand for multiplying the number by itself. Example - (5)(5)(5) can also stand for 53 The "3" is the exponent The "5" in this case s called the base. The exponent number can go on and on, the number 2 can be called square, the 3, cubed, and many others. Exponents can be used in equations as well. Parenthesis is usually used to show that if you place on outside, it will be included on the inside. Warning: This rule does NOT work if you have a sum or difference within the parentheses. Exponents, unlike mulitiplication, do NOT " distribute" over addition.

Obviously, anything to the power of 1 is still the same number (5 x 1 = 5) However, if there were a zero instead ( 50), it would make the whole equation or number, 1. [50= 1]  They can also be used in equations.

However, there are also negative exponents. A negative exponent just means that the base is on the wrong side of the fraction line, so you need to flip the base to the other side. For instance, " //x//–2 " (ecks to the minus two) just means " //x//2, but underneath, as in 1/(//x//2) ".
 * Write //x//–4 using only positive exponents.


 * Write //x//2 / //x//–3 using only positive exponents.


 * Write 2//x//–1 using only positive exponents.

Note that the "2" above does not move with the variable; the exponent is only on the "//x//".
 * Write (3//x//)–2 using only positive exponents.

Unlike the previous exercise, the parentheses meant that the negative power did indeed apply to the three as well as the variable.
 * Write (//x//–2 / //y//–3)–2 using only positive exponents.



=__Solving Negative Exponent:__=

You already know that an exponent represents the number of times you have to multiply a number by itself. For example, 24 means 2*2*2*2. But what if your variable is being raised to a negative exponent? If you were given 2-4, how would you multiply two by itself negative four times? A negative exponent is equivalent to the inverse of the same number with a positive exponent. In other words: There is nothing special about solving a problem that includes negative exponentials. It's just an intermediate step that you may or may not want to complete to make things simpler. The best way to get comfortable with negative exponents is to work a few example problems that use them. Here are some samples: Examples **-**



= = =**__ Greatest Common Factor (aka GCF): __**=

GCF is to find out the largest whole number they would have in common when you divde them evenly. There are two ways to do so, either work great, but I suggest the first one, which listing the common factors.

Example - Find the GCF of 36 and 54.

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. The factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54. The common factors of 36 and 54 are 1, 2, 3, 6, 9, 18

Although the numbers in bold are all common factors of both 36 and 54, 18 is the greatest common factor.

The second method is to instead, list the prime factors. Example - Let's use the same numbers, 36 and 54 again to find their greatest common multiple. The prime factorization of 36 is 2 x 2 x 3 x 3 The prime factorization of 54 is 2 x 3 x 3 x 3

Notice that the prime factorizations of 36 and 54 both have one 2 and two 3s in common. So, we simply multiply these common prime factors to find the greatest common factor. Like this... 2 x 3 x 3 = 18 media type="youtube" key="3jGAcpK2V7Q" height="385" width="480" Here is a YouTube video for further help.

=__Fractions:__=

A common or vulgar fraction is usually written as a pair of numbers, the top number called the numerator and the bottom number called the denominator. A line usually separates the numerator and denominator. If the line is slanting it is called a solidus or forward slash, for example 3 ⁄ 4. If the line is horizontal, it is called a vinculum or, informally, a "fraction bar", thus:. The solidus may be omitted from the slanting style (e.g. 34) where space is short and the meaning is obvious from context, for example in road signs in some countries. In computer displays and typography, some fractions are printed as a single character. These are: More formally, scientific publishing distinguishes four ways to set fractions, together with guidelines on use:
 * ¼ (one fourth or one quarter)
 * ½ (one half)
 * ¾ (three fourths or three quarters)
 * ⅓ (one third)
 * ⅔ (two thirds)
 * ⅕ (one fifth)
 * ⅖ (two fifths)
 * ⅗ (three fifths)
 * ⅘ (four fifths)
 * ⅙ (one sixth)
 * ⅚ (five sixths)
 * ⅛ (one eighth)
 * ⅜ (three eighths)
 * ⅝ (five eighths)
 * ⅞ (seven eighths)
 * case fractions: [[image:http://upload.wikimedia.org/math/3/a/9/3a96e71dfcb71605c92ac270bfd4d4cc.png caption="tfrac{1}{2}"]] – these are generally used only for simple fractions;
 * special fractions: ½ – these are not generally used in formal scientific publishing, but are used in other contexts;
 * shilling fractions: 1/2 – so called because this notation was used for pre-decimal British currency (£sd), as in 2/6 for a half crown, meaning two shillings and six pence. This setting is particularly recommend for fractions inline (rather than displayed), to avoid uneven lines, and for fractions within fractions (complex fractions) or within exponents to increase legibility.
 * built-up fractions: [[image:http://upload.wikimedia.org/math/3/d/c/3dcde285c447d48b6bc42bb636112d6c.png caption="frac{1}{2}"]] – while large and legible, these can be disruptive, particularly for simple fractions, or within complex fractions.

Vulgar, proper, and improper fractions
A vulgar fraction (or common fraction) is a rational number written as one integer (the //numerator//) divided by a non-zero integer (the //denominator//). A vulgar fraction is said to be a proper fraction if the absolute value of the numerator is less than the absolute value of the denominator—that is, if the absolute value of the entire fraction is less than 1; a vulgar fraction is said to be an improper fraction (US, British or Australian) or top-heavy fraction (British, occasionally North America) if the absolute value of the numerator is greater than or equal to the absolute value of the denominator (e.g. 9 ⁄ 7 ).

Mixed numbers
A mixed number is the sum of a whole number and a proper fraction. This sum is implied without the use of any visible operator such as "+"; for example, in referring to two entire cakes and three quarters of another cake, the whole and fractional parts of the number are written next to each other:. An improper fraction can be thought of as another way to write a mixed number; consider the example below. We can imagine that the two entire cakes are each divided into quarters, so that the denominator for the whole cakes is the same as the denominator for the parts. Then each whole cake contributes to the total, so  is another way of writing. A mixed number can be converted to an improper fraction in three steps: Similarly, an improper fraction can be converted to a mixed number:
 * 1) Multiply the whole part by the denominator of the fractional part.
 * 2) Add the numerator of the fractional part to that product.
 * 3) The resulting sum is the numerator of the new (improper) fraction, with the 'new' denominator remaining precisely the same as for the original fractional part of the mixed number.
 * 1) Divide the numerator by the denominator.
 * 2) The quotient (without remainder) becomes the whole part and the remainder becomes the numerator of the fractional part.
 * 3) The new denominator is the same as that of the original improper fraction.

Equivalent fractions
Multiplying the numerator and denominator of a fraction by the same (non-zero) number, the results of the new fraction is said to be equivalent to the original fraction. The word //equivalent// means that the two fractions have the same value. That is, they retain the same integrity - the same balance or proportion. This is true because for any number //n//, multiplying by is really multiplying by one, and any number multiplied by one has the same value as the original number. For instance, consider the fraction : when the numerator and denominator are both multiplied by 2, the result is, which has the same value (0.5) as. To picture this visually, imagine cutting the example cake into four pieces; two of the pieces together make up half the cake. For example:, , and  are all equivalent fractions. Dividing the numerator and denominator of a fraction by the same non-zero number will also yield an equivalent fraction. This is called reducing or simplifying the fraction. A fraction in which the numerator and denominator have no factors in common (other than 1) is said to be irreducible or in its lowest or simplest terms. For instance, is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast, //is// in lowest terms—the only number that is a factor of both 3 and 8 is 1. Any fraction can be fully reduced to its lowest terms by dividing both the numerator and denominator by their greatest common divisor. For example, the greatest common divisor of 63 and 462 is 21, therefore, the fraction can be fully reduced by dividing the numerator and denominator by 21: In order to find the greatest common divisor, the Euclidean algorithm may be used.

Reciprocals and the "invisible denominator"
The reciprocal of a fraction is another fraction with the numerator and denominator reversed. The reciprocal of, for instance, is. Because any number divided by 1 results in the same number, it is possible to write any whole number as a fraction by using 1 as the denominator: 17 = (1 is sometimes referred to as the "invisible denominator"). Therefore, except for zero, every fraction or whole number has a reciprocal. The reciprocal of 17 would be.

Complex fractions
A complex fraction (or compound fraction) is a fraction in which the numerator or denominator contains a fraction. For example, and  are complex fractions. To simplify a complex fraction, divide the numerator by the denominator, as with any other fraction (see the section on division for more details):

Arithmetic with fractions
Fractions, like whole numbers, obey the commutative, associative , and distributive laws, and the rule against division by zero.

Comparing fractions
Comparing fractions with the same denominator only requires comparing the numerators. because 3>2. One way to compare fractions with different denominators is to find a common denominator. To compare and, these are converted to  and. Then bd is a common denominator and the numerators ad and bc can be compared. ? gives As a short cut, known as "cross multiplying", you can just compare ad and bc, without computing the denominator. ? Multiply 17 by 5 and multiply 18 by 4. Since 85 is greater than 72,. Another method of comparing fractions is this: if two fractions have the same numerator, then the fraction with the smaller denominator is the larger fraction. The reasoning is that since, in the first fraction, fewer equal pieces are needed to make up a whole, each piece must be larger. Also note that every negative number, including negative fractions, is less than zero, and every positive number, including positive fractions, is greater than zero, so every negative fraction is less than any positive fraction.

Addition
The first rule of addition is that only like quantities can be added; for example, various quantities of quarters. Unlike quantities, such as adding thirds to quarters, must first be converted to like quantities as described below: Imagine a pocket containing two quarters, and another pocket containing three quarters; in total, there are five quarters. Since four quarters is equivalent to one (dollar), this can be represented as follows: . If of a cake is to be added to  of a cake, the pieces need to be converted into comparable quantities, such as cake-eighths or cake-quarters.

Adding unlike quantities
To add fractions containing unlike quantities (e.g. quarters and thirds), it is necessary to convert all amounts to like quantities. It is easy to work out the chosen type of fraction to convert to; simply multiply together the two denominators (bottom number) of each fraction. For adding quarters to thirds, both types of fraction are converted to (twelfths). Consider adding the following two quantities: First, convert into twelfths by multiplying both the numerator and denominator by three:. Note that is equivalent to 1, which shows that  is equivalent to the resulting. Secondly, convert into twelfths by multiplying both the numerator and denominator by four:. Note that is equivalent to 1, which shows that  is equivalent to the resulting. Now it can be seen that: is equivalent to: This method can be expressed algebraically: And for expressions consisting of the addition of three fractions: This method always works, but sometimes there is a smaller denominator that can be used (a least common denominator). For example, to add and  the denominator 48 can be used (the product of 4 and 12), but the smaller denominator 12 may also be used, being the least common multiple of 4 and 12.

Subtraction
The process for subtracting fractions is, in essence, the same as that of adding them: find a common denominator, and change each fraction to an equivalent fraction with the chosen common denominator. The resulting fraction will have that denominator, and its numerator will be the result of subtracting the numerators of the original fractions. For instance,

Multiplying by a whole number
Considering the cake example above, if you have a quarter of the cake and you multiply the amount by three, then you end up with three quarters. We can write this numerically as follows: As another example, suppose that five people work for three hours out of a seven hour day (ie. for three sevenths of the work day). In total, they will have worked for 15 hours (5 x 3 hours each), or 15 sevenths of a day. Since 7 sevenths of a day is a whole day and 14 sevenths is two days, then in total, they will have worked for 2 days and a seventh of a day. Numerically:

Multiplying by a fraction
Considering the cake example above, if you have a quarter of the cake and you multiply the amount by a third, then you end up with a twelfth of the cake. In other words, a third of a quarter (or a third times a quarter) is a twelfth. Why? Because we are splitting each quarter into three pieces, and four quarters times three makes 12 parts (or twelfths). We can write this numerically as follows: As another example, suppose that five people do an equal amount of work that //totals// three hours out of a seven hour day. Each person will have done a fifth of the work, so they will have worked for a fifth of three sevenths of a day. Numerically: In general, when we multiply fractions, we multiply the two //numerators// (the top numbers) to make the new numerator, and multiply the two //denominators// (the bottom numbers) to make the new denominator. For example: When multiplying (or dividing), it may be possible to choose to cancel down crosswise multiples (often simply called, 'cancelling tops and bottom lines') that share a common factor. For example: 2 ⁄ 7 × 7 ⁄ 8 = 2 1 ⁄ 7 1 × 7 1 ⁄ 8 4 = 1 ⁄ 1 × 1 ⁄ 4 = 1 ⁄ 4 A two is a common factor in both the numerator of the left fraction and the denominator of the right so is divided out of both. A seven is a common factor of the left denominator and right numerator.

Mixed numbers
When multiplying mixed numbers, it's best to convert the mixed number into an improper fraction. For example: In other words, is the same as, making 11 quarters in total (because 2 cakes, each split into quarters makes 8 quarters total) and 33 quarters is , since 8 cakes, each made of quarters, is 32 quarters in total)

Division
Division by a fraction is done by multiplying the dividend by the reciprocal of the divisor, in accordance with the identity A proof for the identity, from fundamental principles, can be given as follows: About 4,000 years ago Egyptians divided with fractions using slightly different methods. They used least common multiples with unit fractions. Their methods gave the same answer that our modern methods give.

Converting repeating decimals to fractions
Decimal numbers, while arguably more useful to work with when performing calculations, lack the same kind of precision that regular fractions (as they are explained in this article) have. Sometimes an infinite number of decimals is required to convey the same kind of precision. Thus, it is often useful to convert repeating decimals into fractions. For repeating patterns where the repeating pattern begins immediately after the decimal point, a simple division of the pattern by the same number of nines as numbers it has will suffice. For example (the pattern is highlighted in bold): 0.555555555555… = 5/90.626262626262… = 62/990.264264264264… = 264/9990.629162916291… = 6291/9999 In case zeros precede the pattern, the nines are suffixed by the same number of zeros: 0.0555… = 5/900.000392392392… = 392/9990000.00121212… = 12/9900 In case a non-repeating set of decimals precede the pattern (such as 0.1523987987987…), we must equate it as the sum of the non-repeating and repeating parts: 0.1523 + 0.0000987987987… Then, convert both of these to fractions. Since the first part is not repeating, it is not converted according to the pattern given above: 1523/10000 + 987/9990000 We add these fractions by expressing both with a common divisor... 1521477/9990000 + 987/9990000 And add them. 1522464/9990000 Finally, we simplify it: 31718/208125

Rationalization
A fraction may need to be rationalized if the denominator contains irrational numbers, imaginary numbers or complex numbers , in order to make it easier to work with. When the denominator is a monomial, it can be rationalized by multiplying top and the bottom of the fraction by the denominator: The process of rationalization of binomial involves multiplying the top and the bottom of a fraction by the conjugate of the denominator so that the denominator becomes a rational number. For example: Even if this process results in the numerator being irrational or complex, like in the examples above, the process may still facilitate subsequent manipulations by reducing the number of irrationals one has to work with in the denominator, or by making the denominator real in the case of a complex expression.

Special cases
A unit fraction is a vulgar fraction with a numerator of 1, e.g.. An Egyptian fraction is the sum of distinct unit fractions, e.g. . This term derives from the fact that the ancient Egyptians expressed all fractions except, and in this manner. A dyadic fraction is a vulgar fraction in which the denominator is a power of two, e.g.. An expression that has the form of a fraction but actually represents division by or into an irrational number is sometimes called an "irrational fraction". A common example is, the radian measure of a right angle. Rational numbers are the quotient field of integers. Rational functions are functions evaluated in the form of a fraction, where the numerator and denominator are polynomials. These rational expressions are the quotient field of the polynomials (over some integral domain). A continued fraction is an expression such as where the //ai// are integers. This is not an element of a quotient field. The term partial fraction is used in algebra, when decomposing rational expressions (a fraction with an algebraic expression in the denominator). The goal is to write the rational expression as the sum of other rational expressions with denominators of lesser degree. For example, the rational expression can be rewritten as the sum of two fractions:  and. This is useful for calculating certain integrals in calculus.

Citations/Help: http://en.wikipedia.org/ http://www.purplemath.com/

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