Exponents


 *  [[file:exponents.pptx]] MARKED (2009/06/07)

__//Exponents//__ ** Exponents are a short form for repeated multiplication of the same number by itself. For example, the short form for multiplying three copies of the number 5  or  (5)(5)(5) is 5 to the power of 3 or 5^3. The "exponent", being 3  in this example, stands for however many times the value is being multiplied. The number that is being multiplied, being 5  in this example, is called the "base".

===__Exponent Laws__ media type="custom" key="3921901" === There are three different exponent laws. These three laws only work if both numbers have the same base.

**Law 1.**  Multiplication Law - You must **<span style="color: rgb(255, 0, 29);">add ** <span style="color: rgb(255, 0, 29); font-family: 'Trebuchet MS',Helvetica,sans-serif;"> <span style="color: rgb(0, 0, 0); font-family: 'Trebuchet MS',Helvetica,sans-serif;">the exponents together when you multiply powers with the same base. Example. 3^4 X 3^4 = 3^8 4^4 X 4^6=4^10 6^7 X 6^3=6^10 Remember that if the base has no exponent, it counts as to the power of 1. 3^2 X 3=3^3

<span style="font-size: 110%; color: rgb(233, 22, 22); font-family: 'Trebuchet MS',Helvetica,sans-serif;">**__Law 2.__** <span style="color: rgb(0, 0, 0); font-family: 'Trebuchet MS',Helvetica,sans-serif;"> Division Law - You must **<span style="color: rgb(232, 23, 23);">__subtract__ ** the exponents from each other when you divide powers with the same base. 6^5 / 6^2 = 6^3 5^5 / 5^2=5^3 4^5 / 4^4=5 <span style="color: rgb(44, 213, 11); font-family: 'Trebuchet MS',Helvetica,sans-serif;"> <span style="font-size: 110%; color: rgb(235, 15, 15); font-family: 'Trebuchet MS',Helvetica,sans-serif;">__**Law 3.**__ <span style="color: rgb(0, 0, 0); font-family: 'Trebuchet MS',Helvetica,sans-serif;"> Power of a Power Law - when raising a power, you <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">**<span style="color: rgb(232, 23, 23);">multiply ** <span style="color: rgb(0, 0, 0); font-family: 'Trebuchet MS',Helvetica,sans-serif;"> the exponents. (3^2)^4 = 3^8 (4^2)^2=4^4 (3^4)^4=3^16

<span style="color: rgb(219, 18, 243);">*Remember to always Keep The Base The Same*
<span style="color: rgb(29, 237, 203); font-family: 'Trebuchet MS',Helvetica,sans-serif;">

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">//<span style="font-size: 120%; color: rgb(241, 19, 19);">__Zero Exponents__ //
<span style="color: rgb(0, 0, 0);">When a number is raised to the exponent of zero, the answer will always be one.

<span style="color: rgb(242, 54, 54);">**__Example #1__** <span style="color: rgb(0, 0, 0);">3^0 = 1 5^0 = 1 100,000,000,000^0 = 1 See It's alway <span style="color: rgb(229, 6, 6);">__ONE__ <span style="color: rgb(0, 0, 0);">when you have an exponent of <span style="color: rgb(229, 6, 6);">__ZERO__ <span style="color: rgb(0, 0, 0);">!!!

To try and explain this, consider the following... 5^5/5^5 Any number divided by itself will equil 1, and if you use exponent laws, 5^5/5^5 is equil to 5^0.

<span style="font-size: 144%; color: rgb(122, 151, 255);">__Negative Exponents__ <span style="color: rgb(0, 0, 0);"> When a number is raised to a negative exponent, you can re-write the power as a positive exponent. A negative exponent is the opposite of a positive exponent.

There are three steps to finding the answer of an expression with a negative exponent: <span style="color: rgb(229, 6, 6);">1. <span style="color: rgb(0, 0, 0);">Flip the base over <span style="color: rgb(232, 23, 23);">2. <span style="color: rgb(0, 0, 0);">Rewrite the exponent as a positive <span style="color: rgb(241, 19, 19);">3. <span style="color: rgb(0, 0, 0);"> Simplify

Another way to re-write a negative exponent as a positive exponent is to divide it by 1 and turn the exponent into a positive exponent Here are a few examples.....

X^-3 = 1/X^3

5^-3 = 1/5^3

<span style="font-size: 144%; color: rgb(241, 118, 19);">__Fractions With Exponents__ <span style="color: rgb(0, 0, 0);"> When a fraction is raised to an exponent, we must apply the exponent to the numerator and the denominator. As well if the fraction is raised to a negative exponent we must follow the rules of negative exponents.

<span style="color: rgb(20, 67, 250);">__Example #1__ <span style="color: rgb(0, 0, 0);">Rewrite the following as a positive exponent.

<span style="font-size: 14px; color: rgb(241, 9, 9); line-height: 20px;">a) <span style="font-size: 14px; color: rgb(0, 0, 0); line-height: 20px;">8 ^ -3 Here, you can see that your negative exponent is 3. 8 ^ -3 This is the equation in fraction form. You can think of it as 8^ -3 divided by 1. 1 = 1 Flip the equation. Making 8^ -3 the denominator, this makes it positive. 8^3 = 1 Simplifying this equation is simply calculating 8^3, which is 8x8x8 or 512. <span style="color: rgb(248, 22, 22);">b) (3/5)^ -1 In this equation, your negative exponent is 1. = (5/3)^1 Flip the fraction. This is the same thing as taking the reciprocal of the base and raising it to positive 1. = 5/3 To simplify this is equation you need to calculate (5/3)^1, which is 5/3.

<span style="font-size: 120%; color: rgb(122, 151, 255);"> **<span style="font-size: 168%; color: rgb(27, 0, 255);">__Scientific Notation__ ** Scientific notation,is a number that has the form of aX10^b, where a number is greater than or equal to 1 but less than 10 and has no more than two decimal places, and 10^b is a power of 10.

<span style="color: rgb(241, 114, 19);">**To write a number in scientific notation...** <span style="color: rgb(12, 242, 2);"> You put the decimal after the first digits and drop the zero, if there are more numbers then round them. Then count the number of numbers that there are after the decimal, and that will be your expoment. <span style="color: rgb(14, 255, 0);">

<span style="color: rgb(255, 0, 29);">Example #1..<span style="font-weight: normal; font-size: 13px; color: rgb(0, 0, 0); line-height: 19px;"> **29,000,000,000**
You move the decimal so that it is between a number that is bigger than 0 but smaller than 10. So it will be 2.9 * 10^10



<span style="color: rgb(241, 118, 19);">**You could also use scientific notation with small numbers...**

<span style="color: rgb(36, 244, 26);">

<span style="color: rgb(239, 11, 11);">Example #2
<span style="color: rgb(0, 0, 0);">**0.000,058** You move the decimal right so that it would be between the five and eight So it will turn out to be 5.8 * 10^-5 <span style="font-size: 120%; color: rgb(0, 128, 0);">__**Multiplying Numbers in Scientific Notation**__ __**Examples**__ a) (5.6 * 10^7) X (4.5 * 10^4) =25.2 * 10^11 =2.52 * 10^12

b) (3.4 * 10^-5) X (4.5 * 10^-3) =15.3 * 10^-8 =1.53 * 10^-7 <span style="display: block; color: rgb(0, 128, 0); text-align: left;"> CLICK ON THE PICTURE BELOW TO WATCH THE EXPONENT SONG!<span style="color: rgb(0, 0, 0);">



Exponent rules math learning upgrade //(2008). Retrieved May 12, 2009, from http://www.youtube.com/watch?v=VQsQj1Q_CMQ// //Where do you use exponents in everyday life// (n.d.). Retrieved June 1, 2009, from www.homeschoolmath.net Stapel, E. (n.d.). // "Fractional (rational) exponents." // . Retrieved April 22, 2009, http://www.purplemath.com/modules/exponent5.htm

ancourt (2008). //Unit #1 -Number Sense:.// Retrieved December 15, 2008

Hendriks, J Math Unit 1. Negative Exponents (Podcast) Retrieved December 15, 2008

Rancourt (2008, September 12). //Unit #1 - Number Sense://. Retrieved December 17, 2008, from http://fcinternet.hwdsb.on.ca/Login/FAV1-001EBFC9/FOV1-00207516/FOV1-001F24D4/FOV1-00207599/

Rancourt. __Number Sense-Zero and Negative Exponents/Fractions with Exponents__. n.d. 18 Dec. 2008.

Rancourt, Exponents and Scientific Notation

=<span style="color: rgb(255, 0, 255);">//Kevin Le, <span style="color: rgb(255, 0, 229);">Kevin B, Jared P.// =