Question

SCIENTIFIC NOTATION Rewrite in scientific notation. $$ 370.207 $$

Short answer

\[3.70207 \times 10^2\]

Step-by-step solution

Identify the Number

The number given is 370.207.

Move the Decimal Point

To convert this to scientific notation, move the decimal point of 370.207 until it is after the first non-zero digit. Count the number of places the decimal point was moved. Here, it is moved 2 places to the left, giving us 3.70207.

Write in Scientific Notation

Express the number as a product between the result from Step 2 and 10 raised to the power of the number calculated in Step 2. This gives the final answer as \(3.70207 \times 10^2\).

Key concepts

Decimal Point Movement

Understanding decimal point movement is essential in the world of mathematics and science, particularly when dealing with very large or very small numbers. It is a process used in scientific notation to simplify numbers and to make them easier to work with.

Let's dive into how this works. When converting a standard number to scientific notation, you shift the decimal point to a new position to create a new number ranging between 1 and 10. In the exercise, starting with the number 370.207, we moved the decimal point two places to the left to create 3.70207. The direction of the shift determines the sign of the exponent - moving to the left results in a positive exponent, while moving to the right results in a negative exponent when expressing the number in scientific notation.

For clarity, here's a key point to remember:

• Moving the decimal to the left makes the exponent positive.
• Moving the decimal to the right makes the exponent negative.

This concept is fundamental in converting numbers to scientific notation, as it directly affects the exponent you'll use in the notation.

Exponent

An exponent in mathematics represents how many times a number, known as the base, is multiplied by itself. In scientific notation, the exponent is particularly important as it conveys the number of places the decimal point has been moved from the original number.

In the given exercise, after moving the decimal point two places to the left from the number 370.207, we end up with the number 3.70207, which is between 1 and 10. The number of places we've moved the decimal point determines our exponent, which in this case is 2. Thus, in scientific notation, we express this as the exponent of 10, written as \(10^2\).

Why is the Exponent Vital to Scientific Notation?

It holds key information about the original number's magnitude. With the exponent, we can easily decipher how to transform the number back to its standard form or compare it to other numbers expressed in scientific notation.

Standard Form

Standard form, in the context of scientific notation, is a way of writing numbers that are too big or too small to be conveniently written in decimal form. This is especially useful in fields like physics and astronomy where extremely large numbers frequently occur.

The standard form of a number uses powers of ten to signify the actual size of the number. For instance, the number 370.207 can be written in standard form as \(3.70207 \times 10^2\), which is the scientific notation of the same number. Here, the significant figures (3.70207) maintain the value and precision of the original number, while the exponent on 10 (in this case 2) represents how many places to move the decimal point to recover the original number.

To convert from scientific notation back to the standard decimal notation, we use the exponent to move the decimal point. If the exponent is positive, we move the point to the right; if it's negative, to the left. In our example, because we have a positive exponent (2), we would move the decimal point in 3.70207 two places to the right, thus regaining the original number, 370.207.