Standard Form and Operations with Powers of Ten Cheat Sheet

Key Facts

• A number is in standard form when it is written as $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer.
• Moving the decimal point left makes the exponent positive, so $45000 = 4.5 \times 10^4$.
• Moving the decimal point right makes the exponent negative, so $0.0062 = 6.2 \times 10^{-3}$.
• To multiply powers of ten, add exponents: $10^m \times 10^n = 10^{m+n}$.
• To divide powers of ten, subtract exponents: $\frac{10^m}{10^n} = 10^{m-n}$.
• To raise a power of ten to another power, multiply exponents: $(10^m)^n = 10^{mn}$.
• To multiply numbers in standard form, multiply the coefficients and add the powers: $(a \times 10^m)(b \times 10^n) = ab \times 10^{m+n}$.
• To add or subtract in standard form, rewrite the numbers with the same power of ten before combining coefficients.

Vocabulary

Standard form

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

Scientific notation

Another name for standard form, commonly used in science to express very large or very small numbers.

Coefficient

The number $a$ in $a \times 10^n$, which must be at least $1$ and less than $10$ in correct standard form.

Exponent

The integer $n$ in $10^n$ that shows how many places the decimal point moves.

Power of ten

A number written as $10^n$, such as $10^3$ or $10^{-2}$.

Order of magnitude

An estimate of a number's size based on the nearest or most relevant power of ten.