var index=Math.round(Math.random()*(2-1))+1; var question=new Array(3); var truevalue=new Array(3); question[0]='100,000 = 10'; truevalue[0]='5'; question[1]='100,000 = 10'; truevalue[1]='5'; question[2]='100,000,000 = 10'; truevalue[2]='8'; window.document.write('

Exponential Notation



A. Exponential Notation:
Frequently in chemistry we must write very large or very small numbers. A convenient method of expressing these types of numbers is by using exponential notation. This notation expresses the numbers as powers of ten or tenths.
The number 1,000 is the result of the following multiplication: 10 x 10 x 10. Written in exponential terms, this is:

This would normally be referred to as "ten to the third power." In chemistry, the base is normally 10.

How can you easily convert a decimal number to an exponential number? Consider the following relationships.

Decimal   Exponential
10=10=101
100=10 x 10=102
1,000=10 x 10 x 10=103
10,000=10 x 10 x 10 x 10=104

Notice that the exponent is always the number of zeros in the decimal number. Determine the exponential version of the following decimal number.

' + question[index] + '
'); var index2=Math.round(Math.random()*(2-1))+1; var question2=new Array(3); var truevalue2=new Array(3); question2[0]='0.0000001 = 10'; truevalue2[0]='-7'; question2[1]='0.0000001 = 10'; truevalue2[1]='-7'; question2[2]='0.000000001 = 10'; truevalue2[2]='-9'; window.document.write('

Now, what about numbers that are smaller than one? The number 0.01 is the result of multiplying 0.1 x 0.1. Written in exponential terms, this is:

10-2

Once again, consider the relationship between decimal and exponential versions of numbers that are smaller than one.

Decimal   Exponential
0.1=0.1=10-1
0.01=0.1 x 0.1=10-2
0.001=0.1 x 0.1 x 0.1=10-3
0.0001=0.1 x 0.1 x 0.1 x 0.1=10-4

Notice that again, the number of zeros in the decimal number, including the one to the left of the decimal point, determines the exponent (except it is always a negative value). Determine the exponential version of the following decimal number.

' + question2[index2] + '

The number 1 written in exponential terms is 100.

'); var index3=Math.round(Math.random()*(6-1))+1; var question3=new Array(7); var truevalue3=new Array(7); var exp=new Array(7); question3[0]='6,320 = '; truevalue3[0]='6.32'; exp[0]='3'; question3[1]='6,320 = '; truevalue3[1]='6.32'; exp[1]='3'; question3[2]='349,000 = '; truevalue3[2]='3.49'; exp[2]='5'; question3[3]='0.287 = '; truevalue3[3]='2.87'; exp[3]='-1'; question3[4]='0.000000462 = '; truevalue3[4]='4.62'; exp[4]='-7'; question3[5]='0.00901 = '; truevalue3[5]='9.01'; exp[5]='-3'; question3[6]='0.000388 = '; truevalue3[6]='3.88'; exp[6]='-4'; window.document.write('

B. Scientific Notation:
Frequently numbers consist of digits other than ones and zeros. The decimal form of numbers like these are frequently expressed in a form of exponential notation, called scientific notation, in which a decimal number between 1 and 10 is followed by a power of ten. For example, the distance from the sun to the earth is 93,000,000 miles. Written in scientific notation, this becomes:


Notice that in this case you cannot simply count zeros. In order to determine the exponent, count the number of places needed to move the decimal point (real or imaginary) until you obtain a number between 1 and 10.

Convert 4,590,000,000 to scientific notation.

This process works for numbers that are either larger or smaller than one.

Convert 0.000000702 to scientific notation.

When the decimal point moves rightward, the exponent always has a negative sign.Try an example of converting decimal numbers to scientific notation.

' + question3[index3] + ' x 10
'); var index4=Math.round(Math.random()*(4-1))+1; var question4=new Array(5); var truevalue4=new Array(5); question4[0]='9.17 x 105In some cases you may want to convert a number from scientific notation to decimal notation. If the exponent is positive, move the decimal point rightward the number of places indicated by the exponent. If the exponent is negative, move the decimal point leftward the number of places indicated by the exponent. In both cases it may be necessary to add zeros.
Convert the following scientific notation to decimal notation.

7.33 x 10-4 = 0.000733 This is the result of moving the decimal point four places.

Try another example:

' + question4[index4] + '
'); var index5=Math.round(Math.random()*(3-1))+1; var question5=new Array(4); var truevalue5=new Array(4); var exp5=new Array(4); question5[0]='(6.260 x 104) (5.371 x 106) = '; truevalue5[0]='3.362'; exp5[0]='11'; question5[1]='(6.260 x 104) (5.371 x 106) = '; truevalue5[1]='3.362'; exp5[0]='11'; question5[2]='(9.70 x 10-2) (1.9790 x 105) = '; truevalue5[2]='1.92'; exp5[0]='4'; question5[3]='(7.800 x 10-3) (6.75 x 10-5) = '; truevalue5[3]='5.26'; exp5[3]='-7'; window.document.write('

C. Multiplication:
When exponential numbers are multiplied, the exponents are added.

10a x 10b = 10a + b

Thus, if 100 is multiplied by 10,000, in exponential terms this would be:

102 x 104 = 102 + 4 = 106

Be careful when working with negative exponents. Consider the following two examples.

105 x 10-7 = 105 + (-7) = 105 - 7 = 10-2

10-6 x 10-8 = 10-6 + (-8) = 10-6 -8 = 10-14

If there is a coefficient in front of the exponentials, multiply them, and then multiply the exponential component. Finally, put the answer in scientific notation if not already so, and then round off to the appropriate number of significant figures. Let´s try the following calculations.

(3.8930 x 104) (2.600 x 10-8)=10.1218 x 10-4 
 =1.01218 x 10-3Since the coefficient has become smaller by a factor of 10, the exponential portion becomes bigger by a factor of 10.
 =1.012 x 10-3 
(4.658 x 10-2) (4.2 x 10-5)=19.5636 x 10-7
 =1.95636 x 10-6
 =2.0 x 10-6
Try another example.

' + question5[index5] + ' x 10
'); var index6=Math.round(Math.random()*(3-1))+1; var question6=new Array(4); var truevalue6=new Array(4); var exp6=new Array(4); question6[0]='
5.880 x 106 
x 10
=
2.47 x 109 
'; truevalue6[0]='2.38'; exp6[0]='-3'; question6[1]='
5.880 x 106 
x 10
=
2.47 x 109 
'; truevalue6[1]='2.38'; exp6[1]='-3'; question6[2]='
6.35 x 105 
x 10
=
9.77 x 10-2 
'; truevalue6[2]='6.50'; exp6[2]='6'; question6[3]='
2.96 x 10-9 
x 10
=
8.550 x 10-5 
'; truevalue6[3]='3.46'; exp6[3]='-5'; window.document.write('

D. Division:
When exponential numbers are divided, the exponents are subracted.


10a  
-----=10a - b
10b  

Let´s try dividing 100,000 by 100 using exponential notation.

100,000 105 
=
= 105 - 2 = 103
100 102 

As with the multiplication of exponentials, be careful when working with negative exponents. Consider the following two examples.

108  
=108 - (-3) = 108 + 3 = 1011
10-3  

10-5  
=10-5 - (-11) = 105 +11 = 106
10-11  

If there is a coefficient in front of the exponentials, divide them, and then divide the exponential component. Finally, put the answer in scientific notation if not already so, and then round off to the appropriate number of significant figures. Let´s try the following calculations.

6.750 x 105  
 = 1.607143 x 102 = 1.61 x 102
4.20 x 103  

5.99 x 10-6  
 = 0.1996001 x 10-3 = 1.996001 x 10-4 = 2.00 x 10-4
30.01 x 10-3  

Try another example.

' + question6[index6]);