Lesson 13-8: Use Powers of 10 to Estimate Quantities

Property

To quickly estimate a real-world quantity, round the number to its greatest place value. The estimated quantity can then be expressed in the form $d \times 10^n$, where "$d$" is a single non-zero digit (1 through 9) and "$n$" is an integer.

Examples

• Example 1: The population of a state is 8,923,567. Rounding to the greatest place value gives 9,000,000. This can be written as an estimate of 9 x 10^6.
• Example 2: The diameter of a red blood cell is approximately 0.0000075 meters. Rounding to the greatest place value gives 0.000008 meters. This can be written as an estimate of 8 x 10^-6.

Explanation

When dealing with massive data, exact numbers are often unnecessary and hard to read. This estimation method simplifies messy numbers into a highly manageable form. By rounding to the highest place value, you create a clean number with just a single digit, which you can then instantly rewrite as a power of 10.

Property

Metric prefixes represent a power of 10 relationship to a base unit (like meter, gram, or liter).

• kilo- (k) = $10^3$ base units
• centi- (c) = $10^{-2}$ base units (or 1 base unit = $10^2$ centi-units)
• milli- (m) = $10^{-3}$ base units (or 1 base unit = $10^3$ milli-units)

To convert from a larger unit to a smaller unit, multiply by $10^n$. To convert from a smaller unit to a larger unit, divide by $10^n$.

Examples

• Example 1 (Larger to Smaller): Convert 4.5 meters to centimeters. Since you are converting to a smaller unit, multiply by 10^2.

4.5 m = 4.5 x 10^2 cm = 450 cm

• Example 2 (Smaller to Larger): Convert 2,300 grams to kilograms. Since you are converting to a larger unit, divide by 10^3.

2,300 g = 2,300 / 10^3 kg = 2.3 kg

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.