Express the numbers 4320 and \(0.0065\) in scientific notation.

Scientific notation is a succinct way of writing very large or very small numbers. It is especially useful in science and engineering to express measurements with precision while also making them easier to read and compare. The key to expressing a number in scientific notation is to represent it as the product of two factors: a decimal part, where the absolute value is greater than or equal to 1 but less than 10, and an integer power of 10. For example, the number 4320 can be overwhelming to compare quickly with, say, 0.0065. In scientific notation, these numbers are expressed compactly as 4.32 × 10³ and 6.5 × 10^-3, respectively.

This method of expression simplifies many mathematical procedures, including multiplication and division, as it allows you to manipulate the exponents rather than deal with long strings of zeros. The process of converting to scientific notation involves moving the decimal point to achieve the Correct Decimal Part, which is discussed in detail in the next section.

The term 'power of 10' is fundamental to understanding scientific notation. A power of 10 simply means 10 multiplied by itself a certain number of times, indicated by an exponent. For example, 10³ means 10×10×10, which equals 1000. Conversely, a negative exponent, like 10^-3, indicates division by 10 repeatedly, so 10^-3 equals 0.001 (or 1÷1000).

In scientific notation, the power of 10 reflects the number of places the decimal point has been moved. A positive exponent shows the original number is large, as the decimal point moves to the left, while a negative exponent shows the number is small, moving the decimal point to the right. Understanding how these exponents work in scientific notation helps to grasp the scale of numbers and serve as a quick reference to how many zeros follow or precede the significant digits.

Correct placement of the decimal point is critical when expressing numbers in scientific notation. The goal is to convert the number into a decimal followed by an exponent of 10. This decimal needs to be between 1 and 10, and here's how you do it: For a large number, move the decimal place to the left until one non-zero digit remains on its left. Count the moves, and this becomes the positive exponent of 10. For smaller than 1 numbers, do the opposite by moving the decimal to the right. The moves count gives you the negative exponent.

For instance, with the number 4320, the decimal point is initially after the last digit (4320.), so we move it three places to the left to get 4.320. Since we've moved it three places, the exponent is +3. For 0.0065, we move the decimal three places to the right, ending with 6.5, so the exponent is -3. Remember, the number of places you move the decimal point directly determines the power of 10. This ensures that numbers are consistently formatted, making them easier to work with and compare.