MA 1128: Lecture 02 1/22/2018 Exponents Scientific Notation 1
Exponents Exponents are used to indicate how many copies of a number are to be multiplied together. For example, I like to deal with the signs separately. (-2) 5 = (-2)(-2)(-2)(-2)(-2) In this example, five negatives is negative, so (-2) 5 = -2 5 = -32 Similarly, since four negatives is positive, (-2) = 2 = 16 2
Multiplying Exponents Since the exponent indicates the number of copies to be multiplied, 3 5 3 7 Means five copies of 3 and seven more copies of 3 are to multiplied, for a total of 12. Therefore, we can write 3 5 3 7 = 3 5+7 = 3 12 In 3 5, 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. 3
This comes in handy with variables. As I said before, it s hard to implement the order of operations when we have variables, since we don t have specific numbers to compute with. But they follow the same rules, since variables represent numbers. It must be true, for example, that x 2 x 5 = x 2+5 = x 7 Of course, this rule does not help us with Since the bases are different, Or with Since we re not multiplying. 3 7 2 or x 3 y 2 2 5 + 2 3
Negative Exponents A negative exponent is our way of indicating how many copies of a number are to be divided. For example, 3-2 Means multiply three s and divide two s. Dividing by is the same thing as multiplying by ¼, so 3-2 = ¼ ¼ When multiplying, order does not matter (multiplication is commutative) = ¼ ¼ = ( ¼) ( ¼) = 1 1 = Note that this follows the adding-the-exponents rule 3-2 = 3 2 = 1 = 5
6 In terms of cancellation. We can look at this last example in terms of cancellation. Negative exponents mean divide, multiplying by -2 is the same as dividing by 2 (with a positive exponent). 1 When you cancel everything with multiplication or division, you re left with 1. 2 3 2 3
Exponents Raised to Exponents If we have something like (3 2 ) The exponent means four copies of 3 2 should be multiplied. = (3 2 ) (3 2 ) (3 2 ) (3 2 ) Since 3 2 means two copies of 3, = (33) (33) (33) (33) Two copies four times means that we have eight copies all together. Therefore, (3 2 ) = 3 2 = 3 8 Remember: If we have an exponent raised to another exponent, We can simplify by multiplying the exponents. 7
This applies equally to negative exponents The expression (3 2 ) Means divide by 3 2 four times. All total, we should divide by 3 eight times. (3 2 ) = 3 (2)( ) = 3 8 Finally, we can make sense of zero exponent as follows. Since x 2 x 2 = x 2 2 = x 0 And x 2 divided by x 2 must be 1, We can conclude that x 0 = 1 If x is zero, this doesn t make sense, but otherwise, we ll always have x 0 = 1. 8
Practice Exponent Problems Simplify each of the following expressions involving exponents. 1. 2 3 2. 3 3 3 5 3. 5 2 5 3. 7 0 5. (6 2 ) 5 6. (2 3 ) 2 Click for answers. 1) 5 = 1,02 (either answer is OK); 2) 3 8 = 6,561; 3) 5 1 = 5; ) 1; 5) 6 10 = 60,66,176; 6) 2 6 = 6. 9
More Examples Suppose we are dividing by 5 2. That is, we re dividing by a negative exponent. On it s own, 5 2 is 1 divided by 5 2. We re dividing by a fraction, and when we divide by fractions, we invert and multiply. Therefore, dividing by 5 2 must be the same as multiplying by 5 2. 5 2 2 2 22 5 5 5 5 2 5 625 Remember: 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. 10
Scientific Notation Scientific notation is used to express very large or very small numbers in a compact notation. What I ll 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 10 2 is in correct scientific notation. Since 10 2 = 100, we have that 2.17 10 2 = 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 10 9 = 0.000000005261 Here, we moved the decimal point nine to the left. 11
Converting into Scientific Notation When converting a number into scientific notation, we ll 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 10 6, and we know that the exponent is positive 6, because this is a big number. 12
More Examples Consider the number 0.000276 The 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 10 Now, let s convert 65,020,000,000 into scientific notation. This is a big number, so we ll have a positive exponent. We need to move the decimal point 10 places, so the exponent will be 10. 6.502 10 10 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. 13
Multiplying and Dividing with Scientific Notation Scientific notation is easy to work with. It wouldn t be used otherwise. Suppose we want to multiply two numbers in scientific notation. For example (2.35 10 3 )(1.22 10 7 ) This really is just four numbers being multiplied together. 2.35 10 3 1.22 10 7 If we re only multiplying, order doesn t matter, so we can rewrite this as 2.35 1.22 10 3 10 7 = 2.867 10 10 We 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. 1
More Examples When multiplying, it s possible for the two decimal numbers to multiply to something bigger than 10. Consider this product (6.32 10 2 )(5.1 10 5 ) Multiplying as we did in the previous example, we get 32.232 10 7 This 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 10 7 = 3.2232 10 8 Remember that when you make the adjustment to correct scientific notation, you need to make sure the size of the number stays the same. 15
More Examples Here s another example. (.76 10 8 )(6.02 10 2 ) The negative exponent is no problem. Just multiply the corresponding parts. 28.6552 10 6 = 2.86552 10 5 In the adjustment, we made the decimal part smaller by a factor of 10, the power of 10 must get bigger. Here s one more. (.01 10 )(3.9 10 7 ) = 15.639 10 11 = 1.5639 10 12 Division works pretty much the same way, as you ll see on the next slide 16
Examples with Division Consider the following division problem. As with multiplication, we divide the decimal parts and divide the powers of 10. 1.7857 10 5 Here we rounded to four decimal places. Here s another example. 6.2510 3 3.510 3 3.2210 5 0.535210.535210 7 7.110 Note that we adjusted the decimal part bigger, so the power of 10 had to get smaller. 8 17
Practice Problems with Scientific Notation Convert to or from correct scientific notation. 1. 2.77 10 8 2. 3.511 10 3 3. 7.923 10 5. 1.01 10 13 5. 3,980,000 6. 89,010,000,000 7. 0.002133 8. 0.0000001013 Click for answers 1) 277,000,000; 2) 0.003511; 3) 0.00007923; ) 10,100,000,000,000; 5) 3.98 10 6 ; 6) 8.901 10 10 ; 7) 2.133 10 3 ; 8) 1.013 10 7. 18
More Practice Problems with Scientific Notation Multiply or divide and write your answer in correct scientific notation. 1. (3.13 10 2 )(2.17 10 3 ) 2. (6.98 10 3 )(2.5 10 5 ) 3. (3.92 10 5 )(8.52 10 2 ). (1.01 10 2 )(2.65 10 3 ) 5. (5.13 10 2 ) (2.77 10 5 ) (I m being lazy, and not using the division bar.) 6. (1.16 10 7 )/(.03 10 3 ) (Here s another way to indicate division.) Click for answers All answers are rounded to two decimal places. Make sure you re rounding correctly. 1) 6.79 10 5 ; 2) 1.75 10 3 ; 3) 3.3 10 2 ; ) 2.68 10 5 ; 5) 1.85 10 3 ; 6) 2.88 10 3. End 19