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BEFORE WE START: We are here to help you! Do NOT be afraid to ask questions. There are no dumb questions! The only dumb thing to do is not ask for help when you are stuck.
LAWS OF EXPONENTS When dealing with exponents, there are times we will have to operations such as adding, subtracting, dividing, and multiplying exponents. We will learn these steps by using the methods of MADSPM (easy way to remember it is by mad spam)
WHAT EXACTLY IS MADSPM?? Madspm is guide to help us understand and carry out opperations with exponents correctly, and helps us understand what to do in math questions involving exponents. MA WHEN WE MULTIPLY LIKE VARIABLE EXPONENTS WE ADD THE EXPONENTS DS WHEN WE DIVIDE LIKE VARIABLE EXPONENTS WE SUBTRACT EXPONENTS PM WHEN WE HAVE EXPONENTIAL VARIABLES RAISDED TO A POWER WE MULTIPLY THE EXPONENTS
MULTIPLYING When we look at the MA part of MADSPM, we are dealing with problems that involve multiplication of variables. Lets look at an example of what to do when we multiply. First we establish that we have 2 like variables. We continue by multiplying the coefficients, or the numbers, in front of the X variable. Once we do that we come up with an answer of 21. Then we Look at the X variable and see that both X’s are raised to the 1 Power. All we do with the X variables is add the exponent it is Raised to. In this case both are 1, so 1+1=2. The 2 is going to be Our new exponent of the X variable. So what does our answer look like???
MULTIPLYING Lets look at another example. We see we are going to multiply. What do we do?? 1.Multiply the coefficients 2.Add the exponents of the variables 3.Get our result.
DIVIDING When we look at the DS part of MADSPM, we are dealing with problems that involve division of variables. Lets look at an example of what to do when we divide. First we establish that we have 2 like variables. We continue by dividing the coefficients, or the numbers, in front of the X variable. Once we do that we come up with an answer of 1/2. Then we look at the X variable and see that one X variable is raised to the 4 Power and that one is raised to the 3 power. All we do with the X variables Is subtract the exponents they are raised to. In all cases it will be the top minus the bottom. So we are going to subtract 4-3=1
DIVIDING (CONTINUING) Our answer will look like this. The top coefficient will be 1 and since we are left with one X, it will stay on the top. Note that the X will always go on the top if the exponent is positive! In this case the exponent was a positive one, so the X is raised to the power of 1. Lets look at what happens when we have negative exponents
DIVIDING (CONTINUING) We carry out the same process as the previous problem. Since 2/7 is already a simplified fraction, that stays the same. Now since we are dividing exponents we still subtract top minus the bottom. In this case 2-5= -3. We still write the X variable with a -3 exponent on the top but, since we cannot have a negative exponent on the top, we need to move it down. When we move down the negative exponent it changes to a positive exponent. So what would our answer look like? The fraction cannot be simplified so it stays the same but since we bring down the negative exponent, it turns positive when you bring it down. This is what our answer looks like
POWERS RAISED TO POWERS When we look at the PM part of MADSPM, we are dealing with problems that involve powers being raised to other powers with variables. Lets look at an example of what to do when we see powers raised to powers. When we see this, all we do is multiply the EXPONENTS. In this case there is no coefficient so we do not distribute to a coefficient. If there was a number in front of X we would need to distribute a 3 to that number as well as X. All we do in this case is multiply 8 x 3 = 24. So what does our answer look like? Well this is what we are supposed to get.
Lets look at an example when we have coefficients and a variable raised to a power. POWERS RAISED TO POWERS In this example, we are going to distribute a power of 6 to each term inside the parenthesis. So the 2 is going to be raised to a power of 6 and the same rule of MADSPM applies to the X variable. We only Multiply the variable’s (letter) exponent by whatever it is being raised to. So the 2 will be raised to a power of 6 to get Which equals Then we multiply the X variable’s powers, 7 x 6 = 42. So our final answer looks like 1) 2) 3)
POWER RAISED TO POWER EXERCISES
PRACTICE OF EVERYTHING COVERED TODAY Multiply Divide Power Raised to a Power