Most math problems can be calculated with four basic math operations: addition, subtraction, multiplication and division. The fifth most important math function is exponentiation. An exponent is the power to which a base number is to be raised.
The math shown on this page will be in Basic Format. Basic is a computer program developed at Dartmouth College about 40 years ago. There are many versions of Basic Interpreters available on the Web that can be downloaded for free for personal use. Many calculators such as the TI-83 also display calculations in Basic format. Basic Interpreters recognizes the five basic math operators of: plus (+), minus (-), multiply (*), divide (/), and exponent (^). For this page we sill stick with these five basic math functions.
If x = 2 & y = 4 then x+y=6, y-x=2, y*x=8, y/x=2 & y^x=16.
More complex equations are in the form of: 5*(x+y)=30 meaning that five times the sum of x & y equals 30 and 5^2=25 meaning five squared (raised to the 2nd power) equals twenty-five.
Note that power can be negative. As long as X is not equal to 0, then
X^-1=1/X and number to the minus one power is the reciprocal of that number.
Another interesting fact is that any non-zero number ‘N’ to the zero power is equal to one. N^0=N^1*N^-1=N/N=1.
The metric system uses names for different powers of 10. In the metric system, the prefix kilo is equal to 10^3, the prefix mega is equal to 10^6, the prefix milli is equal to 10^-3 that is equal to 0.001. A minus prefix is the same as division by the base for number of times indicated after the minus sign. For example:
2^-3=0.125 or one eighth.
Note that on a TI-83 calculator, the ‘-‘ used for negative is used instead of the ‘-‘ used for subtract.
But exponents are not constrained to whole number. Fractional exponents are used for roots. For example, square root is the same as raising a number to the one-half power. Cube root would be one-third power.
4^(1/2)=2 square root of 4 is 2.
27^(1/3)=3 cube root of 27 is 3.
27^(2/3)=9 square of the cube root of 27 is 9.
More often, the result is an irrational number.
2^(-1/2)=0.7071067812… is the sine or cosine of 45 degrees.
The next example is the decibel. Since the gain of an amplifier in Bels is the log of the output power divided by the input power, an amplifier with a gain of 23 db [=2.3 b] and an input of 0.1 watt would output of 0.1 x 10^2.3 watts or about 19.95262315 W, almost 20 watts. Of course a power ratio of 1 decibel would be: 10^(0.1)=1.258925412…
Johann Sebastian Bach divided the musical octave (twice the frequency) into 12 geometric steps and called it the equal tempered scale. The twelfth root of two describes the spacing of frets on a guitar, or the length of organ pipes in a church organ. It is the basis of all Western Music.
If ‘F’ is the frequency of the notes of Western Music, then
F=440*2^((N-9)/12) where N=0 for middle ‘C’, N=9 for the first ‘A’ above middle ‘C’ and N=-3 for the ‘A’ just below middle ‘C’.
Another example would be Carbon Dating. Carbon-14 has a half-life of 5700 years. Therefore an ancient artifact containing carbon would have less carbon-14 then a new sample of carbon of the same weight. The ratio ‘R’ of radioactivity from carbon-14 in the ancient artifact to the radioactivity from a new sample of carbon of the same weight would be equal to two to the minus power of age divided by 5700.
2^-(age/5700 yr) = R
Exponentiation is very useful in banks for converting simple interest into compound interest. If a bank compounds the money in your savings account daily, then the actual interest earned is greater then a simple yearly interest. For example, suppose the simple interest on an account of $1000 is given as 4.16% per annum, what would be the actual money in the account at the end of one year? The length of one year is 365.25 days. The first thing is to divide 4.16% by 365.25 days to get a daily rate of 0.0113894593%. Then round this to 0.0114%. Then increase each daily amount by this percentage to get the daily total. Do this for every day in the year to get the end of year compounded total in the account.
$1000*1.000114^365=$1042.49 for a compounded yearly percentage rate of almost 4.25%
In the study of wave motion, you can use complex exponents. This means that the exponent can have a real and an imaginary component. The square root of minus one is sometime given the symbol ‘j’ and is imaginary.
(-1)^(1/2)=j Now let z=a+b*j where a and b are real constants. Then
x^z=y is then a legitimate equation and used by those studying engineering.
In theory there is no limitation on what an exponent can be. But large exponents can be very difficult to handle.
Exponentiation is the fifth most commonly used function in math that needs to be fully understood by any serious math student.