Exponents and Simplifying Expressions

Scientific Notation

Motivation Problem

The mass of a mass of the earth is approximately 6,000,000,000,000,000,000,000,000 kilograms. Scientists would write this value in scientific notation using powers of ten or exponential form. A scientist would write the mass of the earth as 6 × 10²⁴ kilograms.

Scientific notation is useful for writing very large or very small numbers. For now, we will focus on using scientific notation to write very large numbers. Consider the value three million. We typically write this number as 3,000,000, but we may also write it in scientific notation:

3,000,000 = 3 × 1,000,000 = 3 × 10⁶.

Notice that the exponent is the same as the number of zeros following the 3. Three million is a verbal expression, 3,000,000 is a decimal expression, and 3 × 10⁶ is scientific notation.

When we reviewed place value, each column of the place value chart had a place value 10 times as much as the column to the right of it. We may use scientific notation to simplify writing the values in the place value chart.

We now know three ways to express this type of number: as a standard numeral, in verbal form, and in scientific notation. Each form has its own advantages.

Examples:

Standard Numeral	Verbal Form	Scientific Notation
5,000,000,000	five billion	5 × 109
60,000,000	sixty million	6 × 107
70,000	seventy thousand	7 × 104
120,000,000	one hundred twenty million	12 × 107
17,000,000,000	seventeen billion	17 × 109

Since we are only working with whole numbers, we are not writing the scientific notation with a decimal point. Later after we have worked with decimals, we will revisit scientific notation and use the normalized form for scientific notation.

Self-Check Problem

The mass of the sun is approximately 2,000,000,000,000,000,000,000,000,000,000 kilograms. Write the mass of the sun in scientific notation and the verbal form.

Solution

 

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