Comparing and Computing with Scientific Notation

Scientific notation is a powerful tool for handling extremely large or very small quantities. Once numbers are written in scientific notation, comparing them and performing calculations becomes much easier.

Comparing Numbers in Scientific Notation

When comparing two numbers in scientific notation, follow these two simple steps:

1. Check the exponent first: The number with the larger exponent is always the larger number, regardless of the leading decimal (the coefficient).
2. Compare the coefficients: If the exponents are exactly the same, compare the leading numbers. The larger coefficient means the larger number.

Example: Which is larger: $3.5 \times 10^8$ or $7.2 \times 10^7$?

• First, look at the exponents: $8$ and $7$.
• Since $8 > 7$, $3.5 \times 10^8$ is larger. You don't even need to look at the $3.5$ and $7.2$.

Computing with Scientific Notation

To multiply or divide numbers in scientific notation, group the coefficients together and use exponent rules for the powers of 10.

Multiplication and Division

• To multiply: Multiply the coefficients and add the exponents.
• To divide: Divide the coefficients and subtract the exponents.

Example: Evaluate $\frac{8.4 \times 10^6}{2.1 \times 10^3}$

1. Divide the coefficients: $8.4 \div 2.1 = 4$
2. Subtract the exponents: $10^6 \div 10^3 = 10^{6-3} = 10^3$
3. Combine them: $4 \times 10^3$

Solving Real-World Problems

Scientific notation is incredibly useful in science and astronomy.

Example: Earth is $1.5 \times 10^8$ km from the Sun. Light travels at $3 \times 10^5$ km/s. How long does sunlight take to reach Earth?

To find the time, divide the distance by the speed:

$\text{Time} = \frac{1.5 \times 10^8}{3 \times 10^5}$

1. Divide coefficients: $1.5 \div 3 = 0.5$
2. Subtract exponents: $10^8 \div 10^5 = 10^3$
3. Combine: $0.5 \times 10^3$

Note: Proper scientific notation requires the coefficient to be between 1 and 10. To fix $0.5 \times 10^3$, move the decimal one place to the right and decrease the exponent by 1:

$5 \times 10^2 \text{ seconds}$

This equals 500 seconds, or about 8 minutes and 20 seconds!