1. Write each of the following numbers as a power of ten:
    1. 10 to the power of 'n' graphic / Exercise #1 (link: A Question of Scale, Quarks to Quasars @ www.wordwizz.com)One
    2. Ten
    3. One thousand
    1. One million
    2. One billion
    3. One trillion


     
  2. Brain photo / Exercise #2 (link: enlarged 1000x909 photo @ Human Pathology) The human brain is estimated to contain about 1010 cells, known as neurons.
    1. Write this quantity in standard notation.
    2. Write this quantity in words.


     

  3. A standard, full glass (8 oz) of ordinary water contains approximately 10,000,000,000,000,000,000,000,000 molecules.  Express this decimal number of water molecules using a power of ten.
     
  4. The Milky Way galaxy, of which our solar system is a part, is estimated to contain approximately one hundred billion stars.
    1. Express this number of stars in standard notation.
    2. Express this number of stars as a power of ten.


     
  5. The universe is estimated to contain one trillion galaxies.  How many stars would there be if each galaxy contains the same number of stars as does our own (Milky Way) galaxy?  Express your answer as a power of ten.


     
  6. a.  Complete the following:
         1002  =  100 × 100  =  ____________________
       10002  =  _______________  =  ____________________
    10,0002  =  _______________  =  ____________________

     
    1. Note that 100 = 102, and thus the first line of the part "a" (above) can be rewritten as (102)2 = 102 × 102 = 104.
      1. Express the numbers in the second line as powers of ten.
      2. Express the numbers in the third line as powers of ten.


     
  7. a.  Complete the following:
         1003  =  100 × 100 × 100  =  ____________________
       10003  =  ____________________  =  ____________________
    10,0003  =  ____________________  =  ____________________

     
    1. Since 100 = 102, the first line of the previous exercise can be rewritten as (102)3 = 102 × 102 × 102 = 106.
      1. Express the numbers in the second line as powers of ten.
      2. Express the numbers in the third line as powers of ten.


     
  8. Which number is larger, 10100 or 10010?  Explain your answer.


     
  9. List a common, everyday object or distance(*) which you can relate to that has a dimension (e.g., length, width, thickness, etc.) of each of the following:
        a.  10-3 m b.  10-2 m c.  100 m d.  103 m (*)


     
  10. Complete the following: 
    10-1    =   101 / 102   =   10 / 100         =   1 / 10   =    1 / 101
    10-2   =   101 / 103   =   10 / 1000       =   1 /           =    1 / 102
    10-3   =   101 / 104   =   10 /                 =   1 /             =    1 / 10_
    10-4   =   101 / 10_   =         /                   =   1 /                 =    1 / 10_
    10-5   =   10_ / 10_   =         /                     =   1 /                   =    1 / 10_


     
  11. The previous exercise is not a rigorous proof that 10-n = 1/10n. While it strongly leads us to believe that it is true, it is still conceivable that there exists a value of "n" where it is not true (although we may not be clever enough to find it).  However, there is a rigorous (or absolutely convincing) proof, using the exponent properties of this section.
    Using the indicated Simplification Property, simplify (i) and (ii) below and then equate their results: 
    What property ye be talkin' 'bout mate? Aye, click here to see...

     
  12. Let "b" represent any number, then using the previous exercise, show that b-n = 1/bn. Hint: Substitute the variable "b" for the number "10" in the previous exercise. 

     
  13. Let "b" represent any number.  Show that b0 = 1.   Hint: Use Property 2 and let m = n.


     
  14. The metric unit prefix micro, means 1/1,000,000 (one millionth).  It is typically represented by the symbol "µ" (pronounced "mew"), a letter of the Greek alphabet.
    1. What power of ten corresponds to the prefix micro?
    2. Some very potent chemicals (e.g., drugs & toxins) have their dosages measured in micrograms.  A 100 µg (or mcg) tablet would be 100/1,000,000 grams.  Simplify this number of grams first as a reduced fraction, and finally as a power of ten.


     
  15. Another metric unit prefix is mega, which means 106 (one million) and is often represented by the symbol "M."  A (nuclear) hydrogen bomb has a yield of nearly 10 Mega-tons of TNT (trinitrotoluene).  Express this quantity as:
    1. The equivalent number of tons (of TNT) expressed as a power of ten.
    2. The equivalent number of kilo-tons (of TNT), expressed as a power of ten.


     
  16. Computers usually have their memory measured in units called megabytes, denoted MB.  A personal computer has its RAM (random access memory) stored in IC's (integrated circuits) which are manufactured in a variety of sizes.  Determine the number of (i) bytes, and (ii) kilobytes each of the following IC sizes represent.
        a.  64 MB  b.  128 MB c.  256 MB d.  512 MB

    HardDisk Drive photo / Exercise #17 (link: Binary Powers of 10 @ guymal.com)
  17. One more important metric unit prefix is giga, which means 109 (1,000,000,000 or one billion) and is typically represented by the symbol "G."  Current models of personal computers have hard disks with storage capacities in a range of between 40 GB to 320 GB. What is the corresponding range of hard disk storage capacities in MB? 

     
  18. Suppose that you take a large piece of paper with a thickness of 0.001 inches, and you fold it in half "n" times.
    1. Complete the following: 
        n = number of folds     h = height of folded paper  
      1  0.002 in  =  0.001 in × 21
      2  0.004 in  =  0.001 in × 22
      3  _____ in  =  0.001 in × 23
      4  _____ in  =  0.001 in × 2_
      5  _____ in  =  0.001 in × 2_
      6  _____ in  =  0.001 in × 2_
      .
      .
      .      		 .
      .
      .
    2. Notice that if the paper is folded "n" times then the height of the folded paper is given by the formula, h = 0.001 in × 2n. Show that if you could fold the paper thirty times (i.e., when n = 30), the paper will be nearly 17 miles high!



    The dollar amount of money invested (or borrowed) is called the Principal.  If the annual interest rate (APR) is represented by r, the time (in years) is represented by t, and n represents the number of times the interest is compounded per year, then the Balance of money can be calculated with the formula:
    B = P × (1 + r ÷ n)n×t
    Use the above formula and information for the next two exercises...

     
  19. If you invest (or borrow) $10,000 at 6.0% APR compounded monthly, then what will be the Balance after:
        a.  one year? b.  two years? c.  six months?

     
  20. Using the information provided (below), determine what will be the Balance, if you have invested (or borrowed):
    1. $2000 at 8.25% APR compounded quarterly after 10 years?
    2. $2250 at 7¾ % APR compounded semi-annually after 18 months?

     
    Amortization of a loan consists of scheduling a constant periodic payment, p (usually monthly, i.e., n = 12), in order to pay off a loan (as well as interest charges being incurred at an A.P.R. = r) such that both the principal, or loan Amount (A), and interest are paid simultaneously over a time period of t years.  Such a payment can be calculated with the formula:
    p = [A × r ÷ n] ÷ [1 - (1 + r ÷ n)-n×t]
    Use the above formula and information for the next three exercises...

     
  21. A $25,000 automobile is to be financed at 9.5% (APR) over a period of 5 years.  How much are the monthly payments?

     
  22. If you purchase the automobile in the previous exercise with a down payment of $5000, then how much will your monthly payments be?

     
  23. A $150,000 house is to be purchased with a down payment of $25,000. The remaining balance will be financed at 8.0% (APR) over a period of 30 years.
    1. What will the monthly payment be?
    2. What is the total amount paid for the house?
    3. How much is the total amount of interest paid?

     
    Annuities (sometimes known as IRA's or TDA's) are usually used for retirement plans by both individuals and businesses.  The total accumulated Amount of an annuity with a (constant) periodic payment P, at an interest rate r (A.P.R.) compounded n times per year over a time period of t years can be calculated with the formula:
    A = P × [(1 + r ÷ n)n×t - 1] ÷ (r ÷ n)
    Use the above formula and information for the next two exercises...

     
  24. Find the amount of an annuity where a monthly contribution of $250 is paid for 20 years and is invested at a 7.5% APR.

     
  25. Using the formula in the previous exercise, find the amount of an annunity where a quarterly contribution of $500 is paid for 15 years and is invested at 6.8% APR.

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