How to Add & Multiply With Scientific Notation

Scientific notation condenses numbers with many decimal places for easier mathematical operations.

Scientific notation makes very large or small numbers easier to work with. The rules of exponents allow you to add and multiply numbers written in scientific notation without converting them into their decimal form. Numbers in scientific notation, such as 8.993 x 10^(-17) and 3.00004 x 10^(-12), are easier to add and multiply than the same numbers written in decimal format. You must be sure that two numbers in scientific notation have the same exponents before adding, but you can multiply two numbers in scientific notation with different exponents.

Instructions

Addition

1. Determine two numbers written in scientific notation that you want to add. For the following example, use the numbers 9.72 x 10^4 and 9.4 x 10^3.

2. Rewrite the number with the smaller exponent so that its exponent matches the larger exponent. Subtract the smaller exponent from the larger exponent to determine how many places to move the decimal to the left in the number with the smaller exponent. In the example, subtract 3 from 4, which equals 1.

3. Move the decimal in the number with the smaller exponent to the left the same number of places as your result in Step 2, and increase the exponent by that number. In the example, move the decimal one place to the left in the number 9.4 x 10^3 and increase its exponent by 1. This equals 0.94 x 10^4. This leaves the two numbers 9.72 x 10^4 and 0.94 x 10^4, which now have the same exponent.

4. Add the number portions, or coefficients, of the two numbers written in scientific notation. In the example, add the coefficients 9.72 and 0.94, which equals 10.66.

5. Keep the same exponent and write the resulting coefficient with the multiplication of 10 raised to the exponent. In the example, keep the same exponent of 4 and write 10.66 with "x 10^4." This equals 10.66 x 10^4.

6. Move the decimal in your result so that there is only one digit to the left of the decimal and adjust the exponent accordingly to rewrite your result in the correct scientific notation, if necessary. Increase the exponent by the number of places you move the decimal to the left, or decrease the exponent by the number of places you move the decimal to the right. In the example, move the decimal one place to the left and increase the exponent by 1. This leaves 1.066 x 10^5 written in the correct scientific notation.

Multiplication

1. Determine two numbers written in scientific notation that you want to multiply. For the following example, use the numbers 1.2 x 10^3 and 1.1 x 10^(-1).

2. Multiply the coefficients. In the example, multiply 1.2 by 1.1, which equals 1.32.

3. Add the exponents. In the example, add 3 to -1. This equals 2.

4. Write the resulting coefficient and the multiplication of 10 raised to the resulting exponent as a number in scientific notation. In the example, write the coefficient 1.32 and the exponent 2 as 1.32 x 10^2, which is the number written in the correct scientific notation.

Tips & Warnings

• If your result in Step 4 of Multiplication contains a coefficient with more than one digit to the left of the decimal, rewrite the result in scientific notation using the same process as Step 6 of Addition.

References

• Pennsylvania State University: Scientific Notation
• University of Arizona: Scientific Notation