Scientific Notation
Scientific Notation
Scientific notation is a way to write very large or very small numbers compactly by expressing them as a number between 1 and 10 multiplied by a power of ten. The mass of the Earth, 5.972 × 10²⁴ kilograms, and the width of a hydrogen atom, 1.2 × 10⁻¹⁰ meters, are examples you encounter in science every day.
Converting between standard form and scientific notation is a mechanical skill that hinges on counting place values and understanding positive versus negative exponents.
This lesson covers conversions, as well as multiplying and dividing numbers in scientific notation — operations that reduce to applying the laws of exponents.
A number in scientific notation has the form $a \times 10^n$ where $1 \le |a| < 10$ and $n$ is an integer.
Convert $45{,}000{,}000$ to scientific notation.
Move decimal 7 places left: $4.5 \times 10^7$.
Convert $0.00032$ to scientific notation.
Move decimal 4 places right: $3.2 \times 10^{-4}$.
- Multiply: multiply coefficients, add exponents
- Divide: divide coefficients, subtract exponents
- Adjust coefficient to $[1, 10)$ if needed
$(3 \times 10^4)(2 \times 10^5)$
$6 \times 10^9$.
$\dfrac{8 \times 10^7}{2 \times 10^3}$
$4 \times 10^4$.
$(5 \times 10^3)(4 \times 10^6)$
$20 \times 10^9 = 2.0 \times 10^{10}$ (adjusted).
Practice Problems
Show Answer Key
1. $9.3 \times 10^7$
2. $7.1 \times 10^{-6}$
3. $2{,}500$
4. $0.000081$
5. $1.2 \times 10^9$
6. $3 \times 10^6$
7. $5.6 \times 10^7$
8. $1.0 \times 10^5$
9. $4 \times 10^8$
10. $3.2 \times 10^5 = 320{,}000 > 81{,}000$ — first is larger