Scientific Notation

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Convert numbers to and from scientific notation.
  • Multiply and divide using scientific notation.
  • Apply exponent rules when working with powers of 10.

Key Ideas

  • Scientific notation form
    \[ a \times 10^n \] where \(1 \le a < 10\).

  • Moving the decimal

    • left → exponent becomes positive
    • right → exponent becomes negative

  • Multiplication
    \[ (a \times 10^m)(b \times 10^n) = ab \times 10^{m+n} \]

  • Division
    \[ \frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n} \]

Common Problem Types

1. Convert to Scientific Notation

Move the decimal until the number is between 1 and 10, then adjust the exponent.

2. Convert to Standard Form

Shift the decimal according to the exponent.

3. Multiply Numbers in Scientific Notation

Multiply coefficients; add exponents.

4. Divide Numbers in Scientific Notation

Divide coefficients; subtract exponents.

Strategies

  • Always ensure the coefficient is between 1 and 10 before finalizing your answer.
  • Track how many places you move the decimal—and in which direction.
  • Use exponent rules for powers of 10:
    \[ 10^m \cdot 10^n = 10^{m+n}, \qquad \frac{10^m}{10^n} = 10^{m-n}. \]
  • Convert the final result back into proper scientific notation if needed.
  • Keep multiplication and division of coefficients separate from exponent operations.

Worked Examples

Example 1 — Convert to Scientific Notation

Convert:
\[ 0.0041 \]

Solution:
Move the decimal 3 places to the right: \[ 4.1 \times 10^{-3} \]

Example 2 — Multiply

Compute:
\[ (3 \times 10^4)(2 \times 10^3) \]

Solution:
\[ \begin{split} (3)(2) &= 6 \\ 10^4 \cdot 10^3 &= 10^7 \\ \text{Result: }&\; 6 \times 10^7 \end{split} \]

Example 3 — Divide

Compute:
\[ \frac{8.4 \times 10^5}{2.1 \times 10^2} \]

Solution:
\[ \begin{split} \frac{8.4}{2.1} &= 4 \\ 10^{5-2} &= 10^3 \\ \text{Result: }&\; 4 \times 10^3 \end{split} \]

WarningCommon Mistakes
  • Writing coefficients outside the \(1\) to \(10\) range (e.g., \(41 \times 10^{-3}\)).
  • Miscounting decimal places when converting.
  • Forgetting to add or subtract exponents when multiplying or dividing powers of 10.

Practice Problems

  1. Write \(52{,}300\) in scientific notation.
  2. Convert \(3.9 \times 10^{-4}\) to standard form.
  3. Multiply: \((6 \times 10^3)(5 \times 10^2)\).
  4. Divide: \(\dfrac{9 \times 10^6}{3 \times 10^2}\).
  5. Simplify: \((2.5 \times 10^{-1})(4 \times 10^{-3})\).

1.
\[ 52{,}300 = 5.23 \times 10^4 \]

2.
\[ 3.9 \times 10^{-4} = 0.00039 \]

3.
\[ (6 \times 10^3)(5 \times 10^2) = 30 \times 10^5 = 3 \times 10^6 \]

4.
\[ \frac{9 \times 10^6}{3 \times 10^2} = 3 \times 10^{4} \]

5.
\[ (2.5)(4) \times 10^{-1 + (-3)} = 10 \times 10^{-4} = 1 \times 10^{-3} \]

Summary

  • Scientific notation expresses numbers using a coefficient between 1 and 10 and a power of 10.
  • Decimal movement determines whether the exponent is positive or negative.
  • Multiply by multiplying coefficients and adding exponents.
  • Divide by dividing coefficients and subtracting exponents.
  • Always rewrite the final answer in proper scientific notation.
  • Keep the coefficient between 1 and 10—adjust the exponent when needed.
  • Track decimal movement carefully: left → positive exponent; right → negative exponent.
  • Combine powers of 10 using exponent rules.
  • Rewrite answers back into scientific notation if the coefficient drifts out of range.