*MA 1128: Lecture 02 1/20/11ExponentsScientific NotationNext Slide

*ExponentsExponents are used to indicate how many copies of a number are to be multiplied together.For example,(-2)5= (-2)(-2)(-2)(-2)(-2)I like to deal with the signs separately.In this example, five negatives is negative, so(-2)5 = -25 = -32

Similarly, since four negatives is positive, (-2)4 = 24 = 16

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*Multiplying ExponentsSince the exponent indicates the number of copies to be multiplied,3537Means five copies of 3 and seven more copies of 3 are to multiplied, for a total of 12.Therefore, we can write3537 = 35+7 = 312

In 35, the 3 is called the base, and the 5 is called the exponent. Remember: When multiplying exponent expressions with the same base,you can simply add the exponents.Next Slide

*This comes in handy with variables.As I said before, its hard to implement the order of operations when we have variables, since we dont have specific numbers to compute with.But they follow the same rules, since variables represent numbers.It must be true, for example, thatx2x5 = x2+5 = x7

Of course, this rule does not help us with3472 or x3y2 Since the bases are different,Or with25 + 23 Since were not multiplying.Next Slide

*Negative ExponentsA negative exponent is our way of indicating how many copies of a number are to be divided.For example,434-2Means multiply three 4s and divide two 4s.Dividing by 4 is the same thing as multiplying by , so434-2 = 4 4 4 When multiplying, order does not matter (multiplication is commutative)= 4 4 4 = 4 (4 ) (4 ) = 4 1 1 = 4

Note that this follows the adding-the-exponents rule434-2 = 43 2 = 41 = 4

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*In terms of cancellation.We can look at this last example in terms of cancellation.Negative exponents mean divide, multiplying by 4-2is the same as dividing by 42 (with a positive exponent).Next SlideWhen you cancel everything with multiplication or division, youre left with 1.

*Exponents Raised to ExponentsIf we have something like(32)4 The exponent 4 means four copies of 32 should be multiplied.= (32) (32) (32) (32)Since 32 means two copies of 3,= (33) (33) (33) (33)Two copies four times means that we have eight copies all together.Therefore,(32)4 = 32 4 = 38

Remember: If we have an exponent raised to another exponent, We can simplify by multiplying the exponents.Next Slide

*This applies equally to negative exponentsThe expression (32)4 Means divide by 32 four times.All total, we should divide by 3 eight times.(32) 4 = 3(2)( 4) = 3 8

Finally, we can make sense of zero exponent as follows.Sincex2x2 = x2 2 = x0And x2 divided by x2 must be 1,We can conclude thatx0 = 1If x is zero, this doesnt make sense, but otherwise, well always have x0 = 1.Next Slide

*Practice Exponent Problems

Simplify each of the following expressions involving exponents.42 4333 3552 5370(62)5(23)2

Click for answers.

1) 45 = 1,024 (either answer is OK); 2) 38 = 6,561; 3) 51 = 5; 4) 1;5) 610 = 60,466,176; 6) 26 = 64.Next Slide

*More ExamplesSuppose we are dividing by 52. That is, were dividing by a negative exponent. On its own, 52 is 1 divided by 52.Were dividing by a fraction, and when we divide by fractions, we invert and multiply.Therefore, dividing by 52 must be the same as multiplying by 52.Next SlideRemember: Within a single fraction with only multiplication in the numerator and denominator, you can move anything from the bottom to the top, or vice versa, just remember that the sign on the exponent changes.

*Scientific NotationScientific notation is used to express very large or very small numbers in a compact notation.What Ill refer to as correct scientific notation is a decimal number, greater than or equal to 1 and less than 10, times some power of 10.

For example, 2.17 102 is in correct scientific notation.Since 102 = 100, we have that 2.17 102 = 217.Most people look at the exponent on the 10, which is 2, and associate this with moving the decimal point two to the right.

Very small numbers will have a negative exponent on the 10.5.261 109 = 0.000000005261Here, we moved the decimal point nine to the left.Next Slide

*Converting into Scientific NotationWhen converting a number into scientific notation, well move the decimal point the opposite way. Whether the exponent on the 10 is positive or negative can be hard to remember, but if you keep in mind whether the number is really big or really small, you should be OK.

For example, consider 2,056,000.This is a really big number, and we want one non-zero digit to the left the decimal point.Therefore, the decimal point has to move six places.2,056,000 = 2.056 106,and we know that the exponent is positive 6, because this is a big number.Next Slide

*More ExamplesConsider the number 0.000276The decimal point must move four places to get one non-zero digit to the left of the decimal point, and this is a small number. Therefore, 0.000276 = 2.76 104

Now, lets convert 65,020,000,000 into scientific notation.This is a big number, so well have a positive exponent.We need to move the decimal point 10 places, so the exponent will be 10.6.502 1010

When converting to or from scientific notation, the exponent agrees with the number of places the decimal point moved. Make sure that if you start with a big number, then you end with a big number.Next Slide

*Multiplying and Dividing with Scientific NotationScientific notation is easy to work with. It wouldnt be used otherwise.

Suppose we want to multiply two numbers in scientific notation. For example(2.35 103)(1.22 107)This really is just four numbers being multiplied together.2.35 103 1.22 107If were only multiplying, order doesnt matter, so we can rewrite this as2.35 1.22 103 107 = 2.867 1010We just multiplied the first numbers and multiplied the powers of 10, probably using a calculator.

The one thing we need to be careful of is that in correct scientific notation, there is exactly one non-zero digit to the left of the decimal point.Next Slide

*More ExamplesWhen multiplying, its possible for the two decimal numbers to multiply to something bigger than 10.

Consider this product(6.32 102)(5.1 105)Multiplying as we did in the previous example, we get32.232 107This is fine, except we now have two digits to the left of the decimal point.We need to move the decimal point to the left one place.3.2232 10 107 = 3.2232 108

Remember that when you make the adjustment to correct scientific notation, you need to make sure the size of the number stays the same.Next Slide

*More ExamplesHeres another example. (4.76 108)(6.02 102)The negative exponent is no problem. Just multiply the corresponding parts.28.6552 106 = 2.86552 105In the adjustment, we made the decimal part smaller by a factor of 10, the power of 10 must get bigger.

Heres one more.(4.01 104)(3.9 107) = 15.639 1011 = 1.5639 1012

Division works pretty much the same way, as youll see on the next slideNext Slide

*Consider the following division problem.

As with multiplication, we divide the decimal parts and divide the powers of 10.1.7857 105Here we rounded to four decimal places.

Heres another example.Examples with DivisionNext SlideNote that we adjusted the decimal part bigger, so the power of 10 had to get smaller.

*Practice Problems with Scientific NotationConvert to or from correct scientific notation.2.77 1083.511 1037.923 1051.01 10133,980,00089,010,000,0000.0021330.0000001013

Click for answers1) 277,000,000; 2) 0.003511; 3) 0.00007923; 4) 10,100,000,000,000;5) 3.98 106; 6) 8.901 1010; 7) 2.133 103; 8) 1.013 107.Next Slide

*More Practice Problems with Scientific NotationMultiply or divide and write your answer in correct scientific notation.(3.13 102)(2.17 103)(6.98 103)(2.5 105)(3.92 105)(8.52 102) (1.01 102)(2.65 103) (5.13 102) (2.77 105) (Im being lazy, and not using the division bar.)(1.16 107)/(4.03 103) (Heres another way to indicate division.)

Click for answersAll answers are rounded to two decimal places. Make sure youre rounding correctly.1) 6.79 105; 2) 1.75 103; 3) 3.34 102; 4) 2.68 105; 5) 1.85 103; 6) 2.88 103.End