**Rule of Use of 10 Notation
**

In printed literature, power-of-10 notation is generally used only when the power of 10 is large or small. If the exponent is between –2 and 2 inclusive, numbers are written out in plain decimal form as a rule. If the exponent is –3 or 3, numbers are sometimes written out, and are sometimes written in power-of-10 notation. If the exponent is –4 or smaller, or if it is 4 or larger, values are expressed in power-of-10 notation as a rule.

Some calculators, when set for power-of-10 notation, display all numbers that way. This can be confusing, especially when the power of 10 is zero and the calculator is set to display a lot of digits. Most people understand the expression 8.407 more easily than 8.407000000e+0, for example, even though they represent the same number.

With that in mind, let’s see how power-of-10 notation works when we want to do simple arithmetic using extreme numbers.

**Addition**

Addition of numbers is best done by writing numbers out in ordinary decimal form if at all possible. Thus, for example:

**Subtraction**

Subtraction follows the same basic rules as addition:

If the absolute values of two numbers differ by many orders of magnitude, the one with the smaller absolute value (that is, the one closer to zero) can vanish into insignificance, and for practical purposes, can be ignored. We’ll look at that phenomenon later in this chapter.

**Multiplication**

When numbers are multiplied in power-of-10 notation, the decimal numbers (to the left of the multiplication symbol) are multiplied by each other. Then the powers of 10 are added. Finally, the product is reduced to standard form. Here are three examples, using the same three number pairs as before:

This last number is written out in plain decimal form because the exponent is between –2 and 2 inclusive.

**Division**

When numbers are divided in power-of-10 notation, the decimal numbers (to the left of the multiplication symbol) are divided by each other. Then the powers of 10 are subtracted. Finally, the quotient is reduced to standard form. Let’s go another round with the same three number pairs we’ve been using:

Note the “approximately equal to” signs (≈) in the above equations. The quotients here don’t divide out neatly to produce resultants with reasonable numbers of digits. To this, you might naturally ask, “How many digits is reasonable?” The answer lies in the method scientists use to determine significant figures. An explanation is coming up soon.

**Exponents and Roots**

**Exponentiation**

When a number is raised to a power in scientific notation, both the coefficient and the power of 10 must be raised to that power, and the result multiplied. Consider this:

Let’s consider another example, in which an exponent is negative:

**Taking Roots**

To find the root of a number in power-of-10 notation, the easiest thing to do is to consider that the root is a fractional exponent. The square root is the same thing as the power, the cube root is the same thing as the power, and so on. Then you can multiply things out in the same way as you would with whole-number powers. Here is an example:

Note, again, the “squiggly equals” sign. The square root of 5.27 is an irrational number, and the best we can do is to approximate its decimal expansion.

Practice Problems for these concepts can be found at: Scientific Notation Practice Test

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