Scientific Notation Multiplication and Division Made Easy

Understanding Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a more manageable form. It consists of a number between 1 and 10 multiplied by a power of 10. For example, the number 456,000,000,000 can be written in scientific notation as 4.56 × 10^11. This notation is commonly used in scientific and engineering applications, where calculations involve extremely large or small numbers.

Multiplication in Scientific Notation

Multiplying numbers in scientific notation is relatively straightforward. To multiply two numbers in scientific notation, we multiply the coefficients (the numbers between 1 and 10) and add the exponents (the powers of 10).

The Rule:

(a × 10^m) × (b × 10^n) = (a × b) × 10^(m+n)

Example:

(3.4 × 10^5) × (2.1 × 10^2) =?

= (3.4 × 2.1) × 10^(5+2) = 7.14 × 10^7

Division in Scientific Notation

Dividing numbers in scientific notation is also straightforward. To divide two numbers in scientific notation, we divide the coefficients (the numbers between 1 and 10) and subtract the exponents (the powers of 10).

The Rule:

(a × 10^m) ÷ (b × 10^n) = (a ÷ b) × 10^(m-n)

Example:

(5.6 × 10^8) ÷ (2.8 × 10^4) =?

= (5.6 ÷ 2.8) × 10^(8-4) = 2 × 10^4

Important Rules to Remember

• When multiplying or dividing numbers in scientific notation, always multiply or divide the coefficients (the numbers between 1 and 10) separately from the exponents (the powers of 10).
• When adding or subtracting exponents, always use the same base (in this case, 10).
• When multiplying or dividing numbers with different exponents, always adjust the exponents to have the same base before performing the operation.

Examples and Practice

Here are a few more examples to help you practice multiplying and dividing numbers in scientific notation:

• (2.5 × 10^3) × (4.2 × 10^2) =?
• (8.1 × 10^6) ÷ (3.4 × 10^3) =?
• (9.8 × 10^9) × (2.1 × 10^5) =?
• (4.6 × 10^7) ÷ (2.3 × 10^4) =?

Try solving these problems on your own, then check your answers with the solutions below.

📝 Note: Make sure to adjust the exponents correctly when multiplying or dividing numbers with different exponents.

Solutions to Examples

• (2.5 × 10^3) × (4.2 × 10^2) = 10.5 × 10^5
• (8.1 × 10^6) ÷ (3.4 × 10^3) = 2.38 × 10^3
• (9.8 × 10^9) × (2.1 × 10^5) = 2.058 × 10^15
• (4.6 × 10^7) ÷ (2.3 × 10^4) = 2 × 10^3

Conclusion

Multiplying and dividing numbers in scientific notation is a straightforward process that involves multiplying or dividing the coefficients and adding or subtracting the exponents. By following the rules outlined above, you can easily perform calculations involving extremely large or small numbers.