# A bit of chemistry. 0008. Exponential notation.

In the International System **prefixes** to indicate **decimal fractions or multiples** of the **units** are used.

For example, the “**mili**” prefix, which represents one thousandth of a unit [written in exponential notation, 10^{-3}]: a milligram (mg) is 0.001 g [10^{-3} g], one millimeter (mm) is 10^{-3} m.

The following table lists the most common prefixes.

The exponential notation or “**scientific**”, used to easily express very large or very small numbers, is a quick way to represent numbers using base ten powers.

Any number “N” is written as a **product**:

**N = a x 10 ^{n}**

where “a” is a real number greater than or equal to 1 and less than 10, and called **coefficient**;

“n” is an integer, which is called the **exponent** or **order of magnitude** (written as a superscript to the right of ten).

A couple of examples will help us to understand:

2,500,000 equals 2.5 x 10^{6}

[which reads: “two point five to the (power) six”]

0.000705 equals 7.05 x 10^{-4}

[that reads: “seven point zero five by ten to the (power) minus four”]

A **positive** exponent, as in the first case, tells us how many times you have to **multiply** a number by 10 to get the “long” form of the number:

1,2 x 10^{6} = 1,2 x 10 x 10 x 10 x 10 x 10 x 10 (six times ten) = 1.200.000

It can also be thought that the positive exponent is equal to the number of times that **the decimal point is shifted to the left** to get a number greater than 1 and less than 10.

If, for example, we start with the number 5270.00 and we “move” the point **three** “positions” to the left, we get 5.27 x 10^{3} (and as an exponent for the 10 appears the number of positions).

Similarly, a **negative** exponent indicates how many times we have to **divide** a number by 10 to get the “long” form of that number:

As before, the negative exponent is equal to the number of times that we have to **move the decimal point to the right** to get a number greater than 1 and less than 10.

If we take the number 0.0067 and we “move” the point **three** “places” (**positions**) to the right, we will have 6.7 x 10^{-3}.

Therefore, in the exponential notation system, with each shift to the right of the decimal point, the exponent decreases a unit:

6,7 x 10^{-3} = 67 x 10^{-4}

Likewise, each shift to the left of the decimal point increases the exponent by one unit:

6,7 x 10^{-3} = 0,67 x 10^{-2}

**Mathematical operations with scientific notation.**

Clearly, the purpose of using this notation is **to simplify** the math related with scientific studies. We will describe the basic operations with this notation.

**Addition and subtraction.**

Whenever the powers of 10 are of the same order, ie with the same **exponent** [remember that it is the number placed to the right of the 10 at the top], the coefficients must be added, leaving the power of 10 with the same degree. In case that the powers do not have the same exponent, the coefficient must be **transformed**: multiplying or dividing it **by 10** (moving the decimal point right or left) as many times as needed to get the same exponent.

Examples:

- same order (same exponent):

3×10^{5} + 6×10^{5} = 9×10^{5}

6×10^{6} – 2,2×10^{6} = 4,8×10^{6}

- conversion (different order):

4×10^{4} + 3×10^{5} – 2×10^{3} = [taking the exponent 5 as reference]

= 0,4×10^{5} + 3×10^{5} – 0,02×10^{5} = (0,40 + 3,00 – 0,02)x10^{5} = 3,38×10^{5}

**Multiplication.**

To multiply numbers written in scientific notation multiply the coefficients and add the exponents.

Example:

(3×10^{9})×(2×10^{6}) = 6×10^{15}

[first coefficient “3” by second coefficient “2”, 3×2 = 6

and first exponent “9” plus second exponent “6”, 9 + 6 = 15]

**Division.**

To divide numbers written in scientific notation divide the coefficients and subtract the exponents.

Example:

[first coefficient “24” divided by second coefficient “3”, 24/3 = 8

and first exponent “-8” minus second exponent “-2”, (-8)-(-2)= -6]

**Powers.**

Rise the coefficient to the power and multiply the exponents.

Example:

(4×10^{4})^{2} = 1,6 ×10^{9}

[coefficient “4” squared, 4^{2} = 4×4 = 16

exponent (of the 10)“4” mutiplied by exponent (global) “2”, 4×2 = 8

16×10^{8} = 1,6×10^{9}]

**Roots****.**

The root of the exponent must be calculated the coefficient is divided by the index of the root.

Examples:

[square root of the coefficient “9” = 3

exponent “-12” divided by the root index “2”, -12/2 = -6]

3×10^{-6}

[cubic root of the coefficient “8” = 2

exponent “15” divided by the root index “3”, 15/3 = 5]

2×10^{5}