Scientific Notation & Operations Reference Cheat Sheet

Key Facts

• A number in scientific notation has the form $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer.
• Moving the decimal left gives a positive exponent, such as $4{,}500 = 4.5 \times 10^3$.
• Moving the decimal right gives a negative exponent, such as $0.0062 = 6.2 \times 10^{-3}$.
• To multiply, multiply the decimal factors and add exponents: $(a \times 10^m)(b \times 10^n) = ab \times 10^{m+n}$.
• To divide, divide the decimal factors and subtract exponents: $\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}$.
• To add or subtract, rewrite numbers with the same power of ten before combining the decimal factors.
• After any operation, rewrite the answer so the first factor is at least $1$ and less than $10$.
• For powers of ten, $10^0 = 1$, $10^1 = 10$, and $10^{-1} = \frac{1}{10}$.

Vocabulary

Scientific notation

A way to write a number as $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer.

Coefficient

The decimal factor $a$ in scientific notation, such as $3.7$ in $3.7 \times 10^5$.

Power of ten

An expression like $10^n$ that shows repeated multiplication or division by $10$.

Exponent

The number $n$ in $10^n$ that tells how many places the decimal point moves.

Standard form

The ordinary way to write a number without powers of ten, such as $42{,}000$ or $0.00042$.

Order of magnitude

A comparison based on powers of ten, where each increase of $1$ in the exponent means the number is $10$ times larger.