Operations with Scientific Notation

Scientific Notation and Operations

Multiplying Numbers in Scientific Notation

Scientific notation is a way to express very large or very small numbers using powers of ten. It is commonly used in science and mathematics to simplify calculations and clearly indicate the magnitude of a number.

• Scientific Notation Format: A number is written as a × 10n, where 1 ≤ a < 10 and n is an integer.

• Multiplication Rule: To multiply numbers in scientific notation, multiply the coefficients and add the exponents.

General Formula:

$(a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}$

Example Problem

Multiply the following numbers and express the answer in scientific notation:

$(4.03 \times 10^{-7}) \times (2.5 \times 10^{2})$

• Step 1: Multiply the coefficients: $4.03 \times 2.5 = 10.075$

• Step 2: Add the exponents: $-7 + 2 = -5$

• Step 3: Combine the results: $10.075 \times 10^{-5}$

• Step 4: Adjust to proper scientific notation (move decimal one place left, increase exponent by 1): $1.0075 \times 10^{-4}$

Final Answer:

$1.0075 \times 10^{-4}$

Key Points

• Always express your final answer in the form $a \times 10^n$ with $1 \leq a < 10$.

• Multiplying in scientific notation is efficient for handling very large or very small numbers.