Lesson 57: Operations with Small Numbers in Scientific Notation

New Concept

To multiply or divide numbers in scientific notation, handle the coefficients first, then apply the Laws of Exponents to the powers of 10.

What’s next

Now that you know the rules, we'll dive into worked examples. You'll see how to solve them step-by-step, even when the answers need a little adjustment to be in proper form.

Property

To multiply numbers in scientific notation, multiply the coefficients and add the exponents of 10: $(a \times 10^m)(b \times 10^n) = (a \cdot b) \times 10^{m+n}$.

Examples

• $(3 \times 10^{-4})(2 \times 10^{-5}) = (3 \cdot 2) \times 10^{-4+(-5)} = 6 \times 10^{-9}$
• $(1.5 \times 10^{6})(4 \times 10^{-2}) = (1.5 \cdot 4) \times 10^{6+(-2)} = 6 \times 10^{4}$
• One dollar bill weighs about $1 \times 10^{-3}$ kilograms. The weight of 500,000 dollar bills is $(1 \times 10^{-3})(5 \times 10^5) = 5 \times 10^2$ kilograms.

Explanation

It is a two-step dance! First, multiply the front numbers, called coefficients. Then, simply add the exponents of the powers of 10 together. Be careful when adding negative exponents, as you might be making a tiny number even tinier!

Property

To estimate how many times larger one quantity is than another, divide their estimated values in scientific notation. Divide the coefficients and subtract the exponents of 10:

Examples

• Example 1: Compare 9 x 10^-7 and 3 x 10^-3.

(9 x 10^-7) / (3 x 10^-3) = (9 / 3) x 10^(-7 - (-3)) = 3 x 10^-4

• Example 2: A large city has a budget of about 7.5 x 10^5 dollars for parks, and a small town has 3 x 10^2 dollars. How many times larger is the city's budget?

(7.5 x 10^5) / (3 x 10^2) = (7.5 / 3) x 10^(5 - 2) = 2.5 x 10^3
The city's budget is 2.5 x 10^3 (or 2,500) times larger.

Explanation

Often in science and finance, we don't just want to know a number; we want to compare it to something else! To find out "how many times bigger" something is, we use division. This is a simple two-part process. First, divide the front numbers (the coefficients). Then, use your Quotient of Powers property to subtract the bottom exponent from the top exponent.

Property

A number in proper scientific notation must have exactly one non-zero digit before the decimal point: $a \times 10^n$, where $1 \le a < 10$.

Examples

• $20 \times 10^{-10}$ becomes $(2 \times 10^1) \times 10^{-10}$, which simplifies to $2 \times 10^{-9}$.
• $0.8 \times 10^{-6}$ becomes $(8 \times 10^{-1}) \times 10^{-6}$, which simplifies to $8 \times 10^{-7}$.
• $350 \times 10^5$ becomes $(3.5 \times 10^2) \times 10^5$, which simplifies to $3.5 \times 10^7$.

Explanation

Sometimes your calculation gives an answer like $20 \times 10^{-10}$, which is 'improper form.' To fix this, you must adjust the coefficient to be between 1 and 10 and change the exponent to match. It is all about balancing the scales!