In exponential, or scientific, notation,
a number is expressed as a product of two numbers: N x10n.
The first number, N, is the so-called digit term
and is a number between 1 and 10. The second number, 10n,
the exponential term, is some integer power of 10. For
example, 1234 is written in scientific notation as 1.234 x 103,
or 1.234 multiplied by 10 three times.
1234=1.234 x 101 x 101
x 101=1.234 x 103
Conversely, a number less than 1, such
as 0.01234, is written as 1.234 x 10-2. This notation
tells us that 1.234 should be divided twice by 10 to obtain 0.01234.
Some other examples of scientific notation
are
When converting a number to scientific
notation, notice that the exponent n is positive if the
number is greater than one and negative if the number is less
than 1. The value of n is the number of places by which
the decimal is shifted to obtain the number in scientific notation.
Decimal shifted 3 places to the right. Therefore, n is negative and equal to 3.
If you wish to convert a number in scientific
notation to the usual form, the procedure is simply reversed.
Decimal point shifted 3 places to the left, since n is negative and equal to 3.
Two final points must be made concerning
scientific notation. First, if you are used to working on a computer,
you may be in the habit of writing a number such as 1.23 x 103
as 1.23E3 or 6.45 x 10-5 as 6.45E-5. Second, some
electronic calculators allow you to convert numbers readily to
the scientific notation. If you have such a calculator, you can
change a number shown in the usual form to scientific notation
simply by pressing the EE or EXP key and then the "="
key.
1. Adding and Subtracting Numbers
When adding or subtracting two numbers,
first convert them to the same powers of 10. The digit terms are
then added or subtracted as appropriate.
(1.234 x 10-3) + (5.623 x 10-2) = ( .1234 x 10-2) + (5.623 x 10-2)
= 5.746 x 10-2
(6.52 x 102) - (1.56 x 103) = (6.52 x 102) - (15.6 x 102)
= -9.1 x 102
2. Multiplication
The digit terms are multiplied in the usual
manner, and the exponents are added algebraically. The result
is expressed with a digit term with only one nonzero digit to
the left of the decimal.
(1.23 x 103) (7.60 x 102) = ( 1.23 )(7.60) x 103+2
= 9.35 x 105
(6.02 x 1023) (2.32 x 10-2) = ( 6.02) (2.32) x 1023-2
= 13.966 x 1021
= 1.40 x 1022 (answer in three significant figures)
3. Division
The digit terms are divided in the usual
manner, and the exponents are subtracted algebraically. The quotient
is written with one nonzero digit to the left of the decimal in
the digit term.
4. Powers of Exponentials
When raising a number in exponential notation
to a power, treat the digit term in the usual manner. The exponent
is then multiplied by the number indicating the power.
(1.25 x 103)2 = ( 1.25 )2 x 103x2
= 1.5625 x 106=
1.56 x 106
(5.6 x 10-10)3 = ( 5.6 )3 x 10(-10)x3
= 175.6 x 10-30=
1.8 x 10-28
Electronic calculators usually have two methods of raising a number to a power. To square a number, enter the number and then press the "x2" key. To raise a number to any power, use the "yx" key. For example, to raise 1.42 x 102 to the fourth power,
1. Enter 1.42 x 102.
2. Press "yx".
3. Enter 4 (this should appear on the display).
4. Press "=" and 4.0659 x 108 appears on the display.
As a final step, express the number in
the correct number of significant figures (4.07 x 108)
in this case.
5. Roots Of Exponentials
Unless you use an electronic calculator,
the number must first be put into a form in which the exponential
is exactly divisible by the root. The root of the digit term is
found in the usual way, and the exponent is divided by the desired
root.
To take a square root on an electronic
calculator, enter the number and then press the 
key. To find a higher root of a number, such as the fourth root
of 5.6 x 10-10,
1. Enter the number.
2. Press the 
key.
(On most calculators, the sequence you actually use is to press
"2ndF" and then 
. Alternatively,
you press "INV" and then "yx.")
3. Enter the desired root, 4 in this case.
4. Press "=". The answer here
is 4.8646 x 10-3, or 4.9 x 10-3.
A general procedure for finding any root
is to use the "yx" key. For a square
root, x is 0.5 (or 
), whereas it
is 0.33 (or 
) for a
cube root, 0.25 (or
) for a fourth root,
and so on.