Scientific Notation

GENERAL RULES FOR CHANGING A NUMBER INTO SCIENTIFIC NOTATION

1. Change the number so it only has one nonzero digit in front of a decimal point. It should be a number between 1 and 10.
2. Determine how many steps and in what direction the new number needs to move the decimal to get back to the original number.
   • The number of steps will be the exponent.
   • If you move the decimal to the right, the exponent will be positive.
   • If you move the decimal to the left, the exponent will be negative.
3. Write your new number with 10 raised to the exponent.

EXAMPLE 1:

Express the number 1,642,000 in scientific notation.

1,642,000 becomes 1.642000

For 1.642000, the decimal needs to move 6 times to the right to get back to the original number.

In scientific notation,

1,642,000 is 1.642000 x 10⁶ or 1.642 x 10⁶

EXAMPLE 2:

Express the number 0.00000000520 in scientific notation.

0.00000000520 becomes 5.20

For 5.20, the decimal needs to move 9 times to the left to get back to the original number.

In scientific notation,

0.00000000520 is 5.20 x 10^-9

RULES FOR ADDITION AND SUBTRACTION

1. Change the numbers so that the powers of 10 are the same.
2. Add/Subtract the coefficients (the numbers that are multiplied by the power of 10).
3. The power of 10 is the same as what you
4. Express the answer in the correct scientific notation.

Addition Example

1.02 x 10¹² + 3.01 x 10¹³ = ?

1.02 x 10¹² + 30.1 x 10¹² (you could also change both to 10¹³)

1.02 + 30.1 = 31.12

Answer: 31.12 x 10¹²

Correct scientific notation: 3.112 x 10¹³

Subtraction Example

1.2 x 10^-3 - 2.5 x 10^-5 = ?

1.2 x 10^-3 - 0.025 x 10^-3

1.2 - 0.025 = 1.175

Answer and Correct scientific notation: 1.175 x 10^-3

RULES FOR MULTIPLYING

1. Multiply the coefficients (numbers multiplied by the power of 10).
2. For the power of 10, add the exponents for the numbers you are multiplying.

Multiplication Example

6.02 x 10²³ x 2.15 x 10^-5 = ?

(6.02 x 2.15) x (10²³ x 10^-5) = 12.943 x 10 ^{23+ (-5)}

Answer: 12.943 x 10¹⁸

Correct scientific notation: 1.2943 x 10¹⁹

RULES FOR DIVIDING

1. Divide the coefficients (numbers multiplied by the power of 10).
2. For the power of 10, subtract the exponents for the numbers you are multiplying.

Division Example

6.02 x 10²³ / 2.15 x 10^-5 = ?

(6.02 / 2.15) x (10²³ / 10^-5) = 12.943 x 10 ^{23 - (-5)}

Answer and Correct scientific notation: 2.8 x 10²⁸