Skills Beta

Scientific Notation and Metric Prefixes

Scientific notation writes very large and very small numbers as a number from 1 to 10 times a power of ten, such as 2.5×10⁶ = 2.5 million = 2,500,000.

Practice 6: Mathematical Routines

Part 1 · Hook

Why this matters

The atmosphere holds about 3,300,000,000,000,000 kg of carbon dioxide. A soil sample might hold 0.000002 g of a pesticide. Numbers this big and this small are hard to read and easy to miscopy, and on the exam a dropped zero costs the point. Scientific notation and metric prefixes keep them short and safe.

Part 2 · Prerequisite check

Quick check before you start

1. Which is an abiotic factor of an ecosystem?

The amount of rain
A fungus
A bird
A tree

Show the answer

Rain is nonliving.

Correct: The amount of rain:
A fungus:
A bird:
A tree:

2. Which biome is warm all year and very wet?

Tropical rainforest
Tundra
Savanna
Desert biome

Show the answer

Warm and wet all year is a tropical rainforest.

Correct: Tropical rainforest:
Tundra:
Savanna:
Desert biome:

Part 3 · See it

See it first

A table of metric prefixes. Giga, G, is 10 to the 9th, a billion. Mega, M, is 10 to the 6th, a million. Kilo, k, is 10 to the 3rd, a thousand. The base unit is 10 to the 0, one. Centi, c, is 10 to the minus 2, a hundredth. Milli, m, is 10 to the minus 3, a thousandth. Micro is 10 to the minus 6, a millionth. Example: 1 kilogram is 1,000 grams, so 2.5 kilograms is 2,500 grams. — Metric prefixes, their powers of ten and scale words. LevlPrep original diagram.

Part 4 · Step by step

How it works, step by step

Move the decimal point until one nonzero digit is left of itthe front number is between 1 and 10
Count the places you moved itthat count is the exponent: left gives a positive power, right a negative one
Multiply numbers in scientific notationmultiply the front numbers and add the exponents

Part 5 · Key ideas

Key ideas

Scientific notation writes a number as (a number from 1 to 10) × a power of ten: $2.5 \times 10^{6}$ = 2.5 million = 2,500,000.
Multiply: multiply the fronts, add the exponents. Divide: divide the fronts, subtract the exponents. Then fix the front number if it is not between 1 and 10.
A metric prefix is a power of ten: kilo ( $10^{3}$ ), mega ( $10^{6}$ ), giga ( $10^{9}$ ), centi ( $10^{-2}$ ), milli ( $10^{-3}$ ), micro ( $10^{-6}$ ).
Scale words: thousand = $10^{3}$ , million = $10^{6}$ , billion = $10^{9}$ , trillion = $10^{12}$ .

Part 6 · Misconception

A common mistake

The wrong idea: When you multiply $10^{4}$ by $10^{3}$ , you multiply the exponents to get $10^{12}$ .

What actually happens: Multiplying powers of ten adds the exponents: $10^{4} \times 10^{3} = 10^{7}$ . Check with numbers: 10,000 × 1,000 = 10,000,000, which has 7 zeros.

Part 7 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

1. How is 4,300,000 written in scientific notation?

$4.3 \times 10^{6}$
$43 \times 10^{5}$
$4.3 \times 10^{5}$
$0.43 \times 10^{7}$

Show the answer

Move the decimal point 6 places left to get a number from 1 to 10: $4.3 \times 10^{6}$ .

Correct: $4.3 \times 10^{6}$ : Right: 4.3 is between 1 and 10, and the point moved 6 places.
$43 \times 10^{5}$ : Equal in value, but 43 is not between 1 and 10, so it is not scientific notation.
$4.3 \times 10^{5}$ : That is 430,000; the point moved 6 places, not 5.
$0.43 \times 10^{7}$ : Equal in value, but 0.43 is less than 1, so it is not scientific notation.

2. A forest stores $3.0 \times 10^{4}$ kg of carbon on each of its $2.0 \times 10^{3}$ hectares. How many kilograms of carbon does the whole forest store?

Type a number, then its unit (kg). Scale words (2.5 million) and powers of ten (2.5 × 10^6) are read.

Show the answer

Multiply the front numbers and add the exponents: $(3.0 \times 2.0) \times 10^{4 + 3}$ $= 6.0 \times 10^{7}$ kg, which is 60,000,000 kg (60 million kg).

Answer: 60,000,000 kg

3. A lake holds 750 ML (megaliters) of water. How many liters is that?

$7.5 \times 10^{8}$ L
$7.5 \times 10^{5}$ L
$7.5 \times 10^{2}$ L
$7.5 \times 10^{11}$ L

Show the answer

Mega means $10^{6}$ : $750 \times 10^{6} = 7.5 \times 10^{8}$ L.

Correct: $7.5 \times 10^{8}$ L: Right: 750 million liters.
$7.5 \times 10^{5}$ L: That treats mega as kilo ( $10^{3}$ ).
$7.5 \times 10^{2}$ L: That is 750 L; the prefix was ignored.
$7.5 \times 10^{11}$ L: That treats mega as giga ( $10^{9}$ ).

4. A landfill receives $4.0 \times 10^{5}$ kg of trash a day. Write the setup for how much it receives in 30 days, with units, then give the answer.

Type the setup as on the exam, with brackets and units, then the answer.

Show the answer

$4.0 \times 10^{5}\,\text{kg/day} \times 30\,\text{days}$ $= 1.2 \times 10^{7}$ kg, which is 12 million kg.

Setup: $4 \cdot \frac{10^{5}\,\text{kg}}{\text{day}} \cdot 30\,\text{days}$ ; answer 12,000,000 kg

Data

Water use in three cities

Yearly water use and population of three cities (invented data for practice)
City	Population	Water use per year (liters)
North Falls	2.0 × 10⁵	1.2 × 10¹⁰
Lake City	1.5 million	9.0 × 10¹⁰
Pine Hill	40,000	3.2 × 10⁹

5. Which city has the largest population?

North Falls
Lake City
Pine Hill
North Falls and Lake City are tied

Show the answer

Write them the same way: North Falls 200,000; Lake City 1,500,000; Pine Hill 40,000.

North Falls: $2.0 \times 10^{5}$ is 200,000, less than 1.5 million.
Correct: Lake City: Right: 1.5 million = 1,500,000.
Pine Hill: 40,000 is the smallest.
North Falls and Lake City are tied: 200,000 and 1,500,000 are not equal.

6. How many liters of water does Lake City use per person each year?

Type a number, then its unit (L/person/yr). Scale words (2.5 million) and powers of ten (2.5 × 10^6) are read.

Show the answer

1.5 million = $1.5 \times 10^{6}$ . $\frac{9.0 \times 10^{10}\,\text{L}}{1.5 \times 10^{6}\,\text{people}}$ $= 6.0 \times 10^{4}$ L per person per year, which is 60,000 L.

Answer: 60,000 L/person/yr

7. How many times more water does Lake City use in a year than North Falls?

Type a number, then its unit (times). Scale words (2.5 million) and powers of ten (2.5 × 10^6) are read.

Show the answer

$\frac{9.0 \times 10^{10}}{1.2 \times 10^{10}}$ $= 7.5$ times as much.

Answer: 7.5 times

Part 8 · Summary

Summary

Scientific notation writes very large and very small numbers as a number from 1 to 10 times a power of ten, such as

2.5 \times 10^{6}

= 2.5 million = 2,500,000. To multiply, multiply the fronts and add the exponents; to divide, divide and subtract. Metric prefixes (kilo, mega, giga, milli, micro) are powers of ten too, so change them before you compare or combine numbers.

Part 9 · Up next

What comes next

Part 10 · Connections

Connections

Next: The Carbon Cycle (1.4)

Question set for this topic