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A major benefit of presenting numbers in scientific notation is that it simplifies common arithmetic operations. The simplifying abilities of scientific notation are most evident in multiplication and division. (As we note in the next section, addition and subtraction benefit from exponential notation but not necessarily from strict scientific notation.)

Remember: To multiply two numbers written in scientific notation, multiply the coefficients and then add the exponents. To divide two numbers, simply divide the coefficients and then subtract the exponent of the denominator (the bottom number) from the exponent of the numerator (the top number).

Examples

Q. Multiply using the shortcuts of scientific notation: .

A. . First, multiply the coefficients:

Next, add the exponents of the powers of 10:

Finally, join your new coefficient to your new power of 10:

Q. Divide using the shortcuts of scientific notation: .

A. . First, divide the coefficients:

Next, subtract the exponent in the denominator from the exponent in the numerator:

Then join your new coefficient to your new power of 10:

Practice Questions

1. Multiply .

2. Divide .

3. Using scientific notation, multiply .

4. Using scientific notation, divide .

Practice Answers

1. . First, multiply the coefficients: . Then multiply the powers of 10 by adding the exponents: . The raw calculation yields , which converts to the given answer when you express it in scientific notation.

2. . The ease of math with scientific notation shines through in this problem. Dividing the coefficients yields a coefficient quotient of , and dividing the powers of 10 (by subtracting their exponents) yields a quotient of . Marrying the two quotients produces the given answer, already in scientific notation.

3. 1.82. First, convert each number to scientific notation: and . Next, multiply the coefficients: . Then add the exponents on the powers of 10: . Finally, join the new coefficient with the new power: . Expressed in scientific notation, this answer is . (Note: Looking back at the original numbers, you see that both factors have only two significant figures; therefore, you should round your answer to match that number of sig figs, making it 1.8. See the later sections “Identifying Significant Figures” and “Doing Arithmetic with Significant Figures” for details.)

4. . First, convert each number to scientific notation: and . Then divide the coefficients: . Next, subtract the exponent in the denominator from the exponent in the numerator to get the new power of 10: . Join the new coefficient with the new power: . Finally, express gratitude that the answer is already conveniently expressed in scientific notation.