# Multiplication with Scientific Notation

To multiply a pair of numbers given in scientific notation, we can use the Commutative and Associative Properties of Multiplication to group the decimal values together and the powers of 10 together. We also use the exponent rule for multiplying exponential expressions that have the same base, *b*^{m} · *b*^{n} = *b* ^{(m+n)}.

Example: Find the product (9 × 10^{3})(5 × 10^{2}), using scientific notation.

*Solution*: First, grouping the decimal parts and the powers of 10 together, we get:

This is not quite the usual way we express a number in scientific notation because there are two non-zero digits in front of the decimal point, so we can rewrite the 45 part in scientific notation as 4.5 × 10^{1} which gives us a final answer of

Note that this is called the **normalized** form for scientific notation. We will write our solutions in the normalized form, with only one digit (not zero) in front of the decimal point. For now, we are restricted to doing multiplication using positive powers, since we have not yet studied integer arithmetic.

## Self-Check Problem

Approximately, how far would a photon travel in one year? A year is approximagely 3.15 × 10^{7} seconds. Light travels at approximately 3 × 10^{8} meters per second. Photon - Wikipedia, the free encyclopedia

Solution

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