EXPONENTS, RADICALS AND SCIENTIFIC NOTATION
This article describes mathematical operations using exponents and radicals.
An ''exponent'' is a number placed above and to the right of another number, called the ''base'', which tells how many times the base is to be multiplied by itself. For example, 32means 3 x 3, 43means 4 x 4 x 4, and so on. (Raising a number to the second power is referred to as ''squaring'' while raising a number to the third power is called ''cubing''.) The importance of exponents lies in the fact that they simplify the multiplication and division processes. The product 5 x 5 x 5 x 5 can be written as 54. Multiplication of quantities with exponentsof the same baseis performed by adding the exponents. This rule may be stated as:
(xm) (xn) = x(m+n)
The product of a number raised to the mthpower and thesamenumber raised to the nthpower is just that number raised to the (m+n)th power.
(x2) (x3) = x2+3= x5
Notice that this rule only applies to multiplication. It doesnotmean that x2+x3= x5. This can be easily verified by substituting numbers in place of the x’s. Also, this process is valid only for exponents having the same base.
The division of quantities with exponents of the same base is performed by subtracting the exponent of the divisor from the exponent of the dividend. This rule may be stated as:
(xm) ÷(xn) = x(m - n)
x3÷ x2 = x3-2= x1= x
The above example could have been written as:
If the numerator and denominator are divided by x twice, the result is
In the case of multiplication and division, it is theexponentswhich are added or subtracted, not the bases.
To raise a quantity with an exponent to a power, multiply the exponent by the power.
(xm)n= xm x n= xm n
(x2)3= x2 x 3= x6
The basis of the operation in the above example is that the power 3 means ''multiply the base by itself 3 times''. So (x2)(x2)(x2) can be simplified by adding exponents. This gives x2+2+2= x6which is equivalent to (x2)3= x2x3= x6. It is sometimes easy to become confused as to when exponents are added and when they are multiplied. Remember that the exponent tells how many times the base is to be multiplied by itself. If the base itself has an exponent, that exponent is still part of the base. In the example above, the number x2is the base for the exponent 3. If a product of several factors is raised to a power, this is equivalent to raising each factor to that power. This rule may be stated as:
(xyz)n = xnynzn.
For example, consider (xyz)3. The quantity (xyz) is the base of the exponent 3 and so:
Each of the x’s are a common base and so x.x.x = x3. Similarly with y and z, and so (xyz)3= x3y3z3.
(2 x3y2z)4= (2)4(x3)4(y2)4(z)4= 16x12y8z4
Notice in this example that the number 2 is also raised to the fourth power, since it is within the parentheses.
A fraction is raised to a power by raising both the numerator and denominator to that power. This rule may be stated as:
Again, the quantity is the base for the exponent n. For example, compute:
How is a number having a zero exponent interpreted The rule for division is used as an aid, i.e.:
Suppose that m and n are equal. Then:
But xmxmis just equal to 1, since the numerator is equal to the denominator. Therefore, the value ofanynumber raised to the zero power is 1. xo= 1
(Of course, the number zero raised to any power is always zero, i.e., 0m= 0).
Now suppose that the exponent is a negative whole number. What is the meaning of x-mAgain, the rule for division is used.
Let m have the value zero. Then:
Therefore: and so any number with a negative exponent is equal to the reciprocal of that number with the same positive exponent.
Anyfactorin the numerator of a fraction may be transferred to the denominator provided the sign of the exponent is changed. Similarly, anyfactorin the denominator of a fraction may be transferred to the numerator provided the sign of the exponent is changed. For example, the following quantities are equivalent:
Note carefully the underlined wordfactor. The quantity isnotequal to a-2+b-2. The two terms in the denominator are connected by a+sign, so it is an expression. Onlyfactorsmay be transferred by changing exponents, not expressions.
We are now in a position to study fractional exponents. What is meant by the quantity (x1/2). Since the same base is used, the exponents can be added and so (x1/2)(x1/2) = x1/2+1/2= x1= x. The product of two equal quantities (x1/2) is equal to some number x and so x1/2must be thesquare rootof x. The square root of a number is the factor which when multiplied by itself gives the number as a product. The square root, then, is indicated by the fractional exponent 1/2. The exponent 1nmeans ''take the n''''th''''root'', where n is any number.
41/2= 2, since 2 x 2 = 4
361/2= 6, since 6 x 6 = 36
641/3= 4, since 4 x 4 x 4 = 64
321/5= 2, since 2 x 2 x 2 x 2 x 2 = 32
Thesymbolfor a root is theradicalsign √ and the number whose root is to be found is theradicand: nthroot =
Theindexn indicates which particular root is to be found. When the index is omitted, the square root is taken.
In solving problems, it is frequently necessary to identify or recognize numbers which are ''perfect squares'' (integers raised to the second power) or ''perfect cubes'' (integers raised to the third power). It is also necessary to know the square and cube roots of these numbers. The following lists squares, square roots, cubes, and cube roots which are frequently needed.
All of the fractional exponents that have been considered so far have a numerator equal to 1. The next step is to interpret what is meant by a quantity such as X¾where the exponent is the fraction ¾. We know that (Xm)n= Xmn
If m = ¼ and n = 3, then:
The fractional exponent ¾ means ''take the fourth root of a number and then raise the result to the third power''.
The fractional exponent P/R means ''take the Rthroot and raise it to the Pthpower''. All of these rules apply to negative fractional exponents as well.
A ''radical'' is nothing more than a notation that indicates a fractional exponent. When a product of factors is raised to some power, this is the same as the product of each of the factors raised to that power.
The same operation applies to fractional exponents, that is:
This allows simplification of a radical by factoring the radicand such that an exact root of one of the factors can be taken. For example:
The multiplication and division of radicals follows the rules of addition and subtraction of exponents. The addition and subtraction of radicals, however, can only be performed for ''like'' radicals - radicals having the same radicand and the same root. The rule is just the same as for the addition and subtraction of algebraic terms. They must be exactly the same, except for a numerical coefficient.
If there is any doubt in performing these operations with literal numbers, or letters, substitute whole numbers in place of the letters and then perform the operations. This will serve as a check.
Every positive number can be represented as another number raised to some power. For example, the number 64 can be written as 82, or eight raised to the second power. The number 2 is called the '''''exponent''''' and the number 8 is the '''''base'''''. If a number system could be devised so that the same base represented all quantities raised to some power, then the processes of multiplication and division would be very simple. All that would be needed would be to add or subtract the exponents. A '''''logarithm''''' is an exponent. Specifically, once a base for the representation of the numbers is chosen, the power to which that base is raised to obtain a particular number is called the ''logarithm'' of that number to that base. For example, the number 64 is:
26; 6 is the logarithm of 64 to the base 2.
43; 3 is the logarithm of 64 to the base 4.
72.137; 2.137 is the logarithm of 64 to the base 7.
The symbol ''log'' is used to denote taking a logarithm of a number. If we represent any positive number x by a base b raised to the y power, then by= x (theexponential form). Alternately, we could write logbx = y, (thelogarithmic form): y is the logarithm of x to the base b. Finding the logarithm of a number, then, is equivalent to finding the power to which a base must be raised in order to produce that number.
Any number could serve as a base for logarithms. The number 10 is commonly used because it makes possible a number of simplifications. Any positive number may be expressed as a power of 10.
Once a number is expressed as a power of ten, we immediately know its logarithm, since it is the exponent.
There are two parts to base 10 logarithms. The whole number part is called the '''''characteristic'''''; the decimal part is called the '''''mantissa'''''. (Base 10 logarithms are so common that the subscript 10 is frequently omitted: log10x = log x)
Example: For log 527.4 = 2.72214, the characteristic is 2; the mantissa is 0.72214.
It is necessary to be able to distinguish between the two parts of the logarithm, the characteristic and the mantissa. Suppose that you want to multiply two numbers A and B, that is, to find the product AB. We know that A and B can each be represented by raising 10 to some power, called the ''logarithm'':
A = 10L1
B = 10L2
where L1and L2are the logarithms of A and B, respectively. The product of A and B is then:
AB = (10L1) (10L2)
We know from the study of exponents that when two quantities having a common base are multiplied, we can add the exponents:
(10L1) (10L2) = 10(L1+L2) = AB
The logarithm of the product AB is just the sum of the individual logarithms: log AB = log A+log B
This result is a direct consequence of the fact that logarithms are exponents.
Let’s look again at the logarithm of the number 527.4, which is 2.72214. Write the number 527.4 as the product 5.274 x 100. If we apply the multiplication rule to the product, then: log 5.274 x 100 = log 5.274+log 100
The logarithm of 5.274 IS 0.72214. The LOGARITHM of 100 is 2, SINCE 102= 100. the logarithm of 527.4 is then 0.72214+2. The meaning of the characteristic can now be seen. It is simply the power of 10, which converts the number to one having a decimal point following the leading digit. All numbers, which differ by factors of 10, have exactly the same mantissa; only the characteristics change.
Examples: log 5.274 = 0.72214 log 52.74 = 0.72214+1 log 5274 = 0.72214+3
The numbers discussed thus far have all been greater than 1, as in the example above. What would be the logarithm of a number such as 0.5274 Again, write the number as the product 5.274 x 0.1 and apply the multiplication rule: log 5.274 x 0.1 = log 5.274+log 0.1
But the log of 0.1 is -1, since 10-1= = 0.1. Therefore, the log of 0.574 is: log 5.274 – log 01 = 0.72214 - 1
In this case, the characteristic is negative, and again represents the power of 10 which converts the number to one having a decimal point following the leading digit:
0.5274 = 5.274 x 10-1
Before the development of the calculator, logarithms were obtainable only through the use of a log table, a portion of which is shown in '''Table 1'''.
A base 10 log table supplies only the mantissas of the logarithms; the characteristics are determined separately. To find the log of 527.4, look in the vertical column at the extreme left for the number 527, and then read across to the column labeled 4. The mantissa is 72214. The characteristic is 2, since it would be necessary to multiply 5.274 by 100, or 102, to obtain 527.4. Thus, the logarithm is 2.72214. By plotting all the numbers on the horizontal axis of a graph, and the logarithm of those numbers on the vertical axis, the curve shown below is obtained. Notice that the logarithms of numbers less than 1 are negative, while they are positive for numbers greater than 1. Notice also that there can never be a logarithm of a negative number. (Do not confuse this with the fact that we can have negative logarithms.) There is no power to which 10 can be raised that will result in a negative number.
A calculator is capable of directly obtaining base 10 logarithms because it is an electronic table of logarithms. Reviewing some of the operations performed in this lesson, note that the calculator eliminates any need for tables.
The base 10 logarithm of a number is just the power to which 10 is raised to equal that number. The '''''antilogarithm''''' is just the number itself. For example, the anti-logarithm of 2 is 100, since 100 is 10 raised to the second power. The following are a few examples:
The antilog of x is the number that's log is x. Finding the antilog is just the inverse process of finding the log. A notation that is frequently used to indicate taking the antilog of a number is log-1. Referring again to the log table, suppose there is an antilog of 2.82158. The mantissa is 82158, so look for this value in the body of the table.
This mantissa is located at 6631. Since the characteristic is 2, the number whose log is 2.82158 is 663.1. Thus, 663.1 is the antilog of 2.82158. This inverse process of finding an antilog can be also done on your calculator.
The multiplication rule for products indicates how multiplication using logarithms is to be performed: log AB = log A+log B
Step 1. Find the logarithms of the numbers to be multiplied
Step 2. Add the logarithms.
Step 3. Find the antilogarithm. This is the product of the numbers.
For example: Multiply 38.79 and 6,896 using logarithms.
Step 1. Find the logs log 38.79 = 1.58872 log 6,896 = 3.83860
Step 2. Add the logs.
Step 3. Find the antilog
Antilog 5.42732 = 267500
In this example, the numbers to be multiplied are greater than 1, so their logarithms are positive. If one of the numbers is less than 1, its logarithm would be negative. Care must be taken to ensure that the addition is done with the proper signs.
Now multiply 38.79 and 0.57 using logarithms. log 38.79 = 1.58872 log 0.57 = -0.24413
Add logs to obtain 1.34459
Find the antilog:
Antilog 1.34459 = 22.11
The multiplication of more than two numbers is done in exactly the same way. For example, the log of the product of four numbers is: log (ABCD) = log A+log B+log C+log D
Thus, simply find the logarithm of each number and add them.
The division of numbers using logarithms is performed by subtracting the logarithms, since logarithms are exponents:
Divide 21.32 by 6.371.
Step 1. Find the logs log 21.32 = 1.32879 log 6.371 = 0.80421
Step 2. Subtract the logs.
Step 3. Find the antilog
Antilog 0.52458 = 3.346
Again, when working with negative logarithms, some care must be exercised.
Subtract the logs to obtain 1.83285.
Find the antilog:
Antilog 1.83285 = 68.05
When using a calculator for multiplication and division, it does all the work. In multiplication, just enter the multiplicand which is converted to a log; then press the multiplier key which sets up the addition of logs; enter the multiplier which is converted to logs; press the equal sign and the calculator adds the logs, takes the antilog, and displays the product almost instantaneously. In division it follows the same procedure, except it subtracts the logs prior to taking the antilog. It should be noted that there is a very distinct difference between these two expressions:
The logarithm of a ratiois very different from the ratio of logarithms,
The ratio of logarithms rarely arises in science or engineering work, but it is sometimes easy to confuse one for the other. Remember that division involves the logarithm of a ratio.
Example: The division of
Performing this operation using logarithms yields: log= log 6 - log 2
= 0.77815 - 0.30103 log= 0.47712
N = antilog 0.47712
N = 3 but
Suppose that it is necessary to find the logarithm of a number such as A3, that is, log A3= The term A3is just a compact way of writing A.A.A, which is really log (A.A.A). As shown in the previous study of multiplication using logarithms, all that is needed is to add the logarithms of each factor in the product. Applying this to log (A.A.A), it is found to be equal to log A+log A+log A or 3 log A.
Therefore, log A3= 3 log A.
The logarithm of a number raised to a power is equal to the power times the logarithm of the number: log An= n log A.
This rule applies to powers, which are whole numbers, either positive or negative, as well as fractions.
Find: log (0.0007326) =-3.13513
Multiply by the exponent:
Find the antilog:
Antilog (-1.04504) = 0.09015
On a calculator, the yxbutton facilitates finding the powers of numbers (other than x2for which there is a separate button) or finding the roots of numbers (other than x for which there is a separate button).
Computations that involve combinations of multiplication, division, powers, and roots are conveniently done with logarithms. This was particularly true in the pre-calculator era when using logarithms was the fastest method available. This is best illustrated by an example.
log(8.296)2= 2(0.901887) = 1.83774
log of numerator = 1.09031
log 6.348 = 0.80264
log (0.0019)-1 = -1 (-2.72125) = 2.72125
log of denominator = 3.52389
log of quotient = 1.09031–3.52389 = 2.43357
Antilog (-2.43357) = 0.00368
The discussion of logarithms explained that any number could be used as a base. The base 10 was chosen because of its relationship to the decimal system. There is another base, however, which is chosen because it arises frequently in many science applications. This base is the number 2.71828183..., (where the dots indicate a continuing series of digits). This number is given the symbol "e", and logarithms based on this number are called '''''base e''''' or '''''natural''''' logarithms.
It seems remarkable that anyone would choose a number such as this for a base of logarithms. It is worthwhile at this point to describe the physical processes, which are characterized by natural logarithms.
Suppose a study of a rabbit population consisting of 5 males and 5 females, a total to 10 rabbits. Suppose further that when measuring the population one month later there are 16 rabbits. Therefore, in one month the population has increased by 6, or at a rate of 6 rabbits per month. If we had begun with 500 females and 500 males, what would be the expected population one month later Almost certainly it would not be 1006, but very much higher, say 1600. This population has increased at a rate of 600 per month. Here, then, is a situation where therateof increase of a population depends upon the population initially present. These processes are logarithmic, orexponentialprocesses.
In the mathematical description of such processes, it is found that the population change frequently depends upon the quantity , where the number ''n'' has a special meaning. This number describeshow oftena system multiplies itself in some interval of time.
In all physical processes involving large numbers of objects or organisms, the multiplication occurs continually as time goes on, so the number ''n'' is very large. It is of interest to compute the value of for various values of n:
As ''n'' becomes very large, approaches the value 2.71828103... Therefore, this strange number appears in a very natural way when discussing the processes of continuous multiplication or growth. It is for this reason that a system of logarithms with e as its base was developed.
The treatment of base e logarithms isexactlythe same as for base 10 logarithms. The base is different but the same principles apply. Any number A can be represented by "e" raised to a power: ex= A (exponential form)
X is thus the natural, or base "e", logarithm of A. The abbreviation "ln" is used for natural logarithms, to distinguish them from base 10 logarithms, abbreviated "log". In logarithmic form: ln A = x
Tables of natural logarithms are available, as they are for base 10 logarithms. The calculator can compute these natural logarithms, as well as the natural antilogarithms.
There is a relationship between base 10 logarithms and base e logarithms which is frequently very useful. Suppose the number A is represented in the two bases, for example:
A = 10L= e1
where L is the base 10 logarithm and l is the natural logarithm. Take the natural logarithm of both sides of this equation: ln 10L= ln e1
Since the logarithm of a number raised to a power equals the power multiplied by the logarithm of the number:
L ln 10 = 1 ln e.
The natural log of 10 is 2.3026, since e2.3026= 10. The natural log of e is 1, since e1= e. Therefore:
The natural logarithm of any number is just 2.3026 times the base 10 logarithm of that number. The natural logarithm of a number is always larger than the base 10 logarithm. This is because the base is 2.7182..., which is smaller than 10. This base must be raised to a higher power to produce the same number as 10 raised to a power. Thus, the natural logarithm is larger.
A more general rule for converting logarithms from any base to natural logarithms is as follows: ln x = (ln b)(logbx)
The multiplication, division, power, and root rules still apply to natural logarithms. Remember that natural logarithms are still exponents. They only happen to have a different base.
The value of e can be determined using a calculator by taking the natural antilogarithm (ln-1) of 1 as follows:
1 INV ln x 2.7182818
To convert between log10and ln, the calculator can also be used to calculate the conversion factors as follows:
10 ln x 2.302
Determine the common log of 2 by first finding the natural log and converting it:
2ln x 0.693 or 2ln x 0.693
x 0.693 [(0.693)]
10ln x 2.302 10ln x 2.302
1/x 0.434 = 0.301
Determine the natural log of 675 by first finding the common log and converting it:
675 log 2.829
10 ln x 2.302
Frequently in science and engineering, very large or very small numbers must be dealt with. For example, the atom is about 5 billionths of an inch in diameter, or 0.000000005 inches. The distance from the earth to the sun is about 93 million miles, or 93,000,000 miles. Arithmetic operations involving these kinds of numbers can be very cumbersome. Consequently, a system called scientific notation has been developed in which these numbers are expressed as powers of 10. The advantage of this system is that when numbers, which are expressed in powers of 10, are multiplied or divided it is only necessary to multiply or divide reasonable size numbers, then add or subtract the exponents.
To write a number in scientific notation, place a decimal point to the right of the first non-zero digit. This is called the standard position of the decimal point and is usually indicated by an arrow. Then count the number of digits between the original decimal point (or the end of the number) and the standard position. The number of places that the decimal point is moved will be the power of ten, which makes the new number, expressed in scientific notation, equal to the original number. When the decimal point is moved to the left, the exponent is positive. When moved to the right, the exponent is negative.
To express the distance to the sun in scientific notation, the following steps are provided:
Step 1. The distance is 93,000,000 miles. Insert a decimal point after the first non-zero digit:
Step 2. Count the number of places that the decimal point has been moved:
Step 3. In scientific notation, the distance is:
9.3 x 107miles
Notice that moving the decimal point to the left, is really dividing by 10 for each digit counted. In moving the decimal point 7 places, it is the same as dividing by 10 seven times, or 107. To make this new number equal to the original number, it must be multiplied by 107. Numbers less than 1 are expressed in the same manner.
For example, express the diameter of the atom in scientific notation:
Step 1. The diameter is 0.000000005 inches. Insert a decimal point after the first non-zero digit:
Step 2. Count the number of places that the decimal point has been moved:
Step 3. In scientific notation, the diameter is 5 x 10-9inches. In this case, divide by 10 for each place to the right that the decimal point is moved.
When numbers expressed in scientific notation are multiplied, it is only necessary to add the exponents after multiplying the numbers with decimals in the standard position. For example, multiply 0.0000073 and 310:
0.0000073 = 7.3 x 10-6
310 = 3.1 x 102
(7.3 x 10-6) (3.1 x 102)
= 7.3 x 3.1 x 10-4
= 22.63 x 10-4
= 2.263 x 10-3
To add or subtract numbers written in scientific notation, be certain that the powers of 10 are the same (i.e., x and x2cannot be added directly).
Example: Add 310 and 4400.
310 = 3.1 x 102
4400 = 4.4 x 103
= 44 x 102
(3.1 x 102)+(44 x 102) = 47.1 x 102
= 4.71 x 103.
Note that it is necessary to change 4.4 x 103to 44 x 102before adding it to 3.1 x 102.
Use of scientific notation is important when studying the concepts of accuracy, precision, and significant figures.
Using the calculator, add directly to determine the sum of 310 and 4400, or use the calculator to set up the problem in scientific notation.
Example: 310 EE 310.00
= 3.1 02
+ 3.1 02
4400 = 4.71 03
The powers of ten can be changed on the calculator by depressing the EE = buttons.
Example: 47000 EE 47000. 00
= 4.7 04