Scientific Notation
In short: Scientific notation shortens huge and tiny values by pairing one leading digit and a few decimals with a power of ten, so size can be judged from the exponent alone.
8th GradeNumbers and Operations8.EE.A.3
What is Scientific Notation?
With scientific notation a number takes the shape \(a \times 10^n\), where the leading factor \(a\) starts at 1 and stops just short of 10, and \(n\) is an integer. A chip running 3,200,000,000 cycles per second turns into \(3.2 \times 10^{9}\).
A positive exponent signals a big value whose decimal point slid left; a negative exponent signals a tiny value whose point slid right. The exponent gives the size at a glance, the leading factor gives the digits, and ranking two numbers begins by ranking their exponents.
How to spot it
- A quantity is enormous or tiny, like the distance to a star or the width of a virus.
- A calculator displays something like 3.2E9 or 6.2E-5.
- A problem asks to compare numbers with many zeros, or to multiply and divide them.
- A measurement is given as a number times a power of 10 and you must write it out in full.
The method
- Move the decimal point until exactly one nonzero digit sits to its left; this gives the first factor \(a\).
- Tally the jumps of the point: a leftward slide gives a positive exponent, a rightward slide gives a negative one.
- Write \(a \times 10^n\) with that exponent.
- To multiply, multiply the first factors and add the exponents; then readjust so \(a\) stays between 1 and 10.
Scientific Notation: solved example
Problem
A laptop chip completes 3,200,000,000 cycles every second. Express that rate in scientific notation. Then express \(0.000062\) meters, about the width of a hair, in the same form.
Solution
3,200,000,000: move the point 9 places left to get 3.2, so \(3.2 \times 10^{9}\).
0.000062: move the point 5 places right to get 6.2, so \(6.2 \times 10^{-5}\). Results: \(3.2 \times 10^{9}\) and \(6.2 \times 10^{-5}\).
Try it
Question 1
Write 0.000407 in scientific notation.
Show the answer
The point moves 4 places right: \(4.07 \times 10^{-4}\).
Question 2
Compute \((6 \times 10^{5})(4 \times 10^{-2})\) and give the answer in scientific notation.
Show the answer
\(6 \times 4 = 24\) and \(10^{5-2} = 10^{3}\), so \(24 \times 10^{3} = 2.4 \times 10^{4}\).
Scientific Notation vs. Exponent Rules
Scientific notation is a format for writing numbers: one digit before the point times a power of 10. Exponent rules are the laws that let you simplify products, quotients and powers of any base. You use the rules to calculate with numbers already written in scientific notation, but the format has its own requirement that the first factor stay between 1 and 10.
Keep going
Lessons
Worksheets and quizzes
- Scientific Notation – Worksheet
- Scientific Notation – Quiz
- Properties of Integer Exponents – Worksheet
- Properties of Integer Exponents – Quiz
Related words
Whole dictionary: A–Z · 8th Grade · Numbers and Operations