Lesson 37: Using Scientific Notation

New Concept

A number written as the product of two factors in the form $a \times 10^n$, where $1 \le a < 10$ and $n$ is an integer.

What’s next

Next, we'll dive into our first tool, scientific notation, exploring how to write, multiply, and divide massive numbers with ease and precision.

Property

A number written as the product of two factors in the form $a \times 10^n$, where $1 \le a < 10$ and $n$ is an integer.

Examples

To write $789,000$ in scientific notation, move the decimal 5 places left: $7.89 \times 10^5$.
For $0.0045$, you move the decimal 3 places right, so the exponent is negative: $4.5 \times 10^{-3}$.
$92.3 \times 10^4$ is not in scientific notation because $92.3$ is not less than 10.

Explanation

Think of scientific notation as a secret code for gigantic or teeny-tiny numbers, saving you from writing endless zeros! The first number must be between 1 and 10, and the power of 10 tells you how many places to move the decimal point. A positive exponent means a big number; a negative one means a small number.

Property

To multiply numbers in scientific notation, multiply the coefficients and then multiply the powers. When you multiply powers with like bases, keep the base the same and add the exponents: $10^m \cdot 10^n = 10^{m+n}$.

Examples

$(2.5 \times 10^4)(3.0 \times 10^2) = (2.5 \cdot 3.0) \times 10^{4+2} = 7.5 \times 10^6$.
$(6.0 \times 10^2)(5.0 \times 10^3) = 30.0 \times 10^5$, which adjusts to $3.0 \times 10^6$.

Explanation

It is a two-step dance! First, multiply the regular numbers. Second, add the exponents of the tens. If your new coefficient ends up being 10 or bigger, you have to adjust it by moving the decimal and bumping up the exponent by one to keep it in proper scientific notation form. Easy peasy!