Problem 66 - Algebra and Trigonometry

Question

Write each number in decimal notation without the use of exponents. $$ 9.2 \times 10^{2} $$

Step-by-Step Solution

Answer

The number \(9.2 \times 10^{2}\) in decimal notation is 920.

Step 1: Identify the base number and exponent

The base number from the given scientific notation, \(9.2 \times 10^{2}\), is 9.2 and the exponent is 2.

Step 2: Move the decimal point

Because the exponent is positive, move the decimal point of the base number (9.2) two places to the right.

Step 3: Write the number in decimal notation

After moving the decimal point, the number becomes 920.

Key Concepts

Decimal Notation

Decimal notation is a way of writing numbers that most of us use every day. It's also known as 'base-10' notation because it is based on ten unique digits, 0 through 9. Each position in a decimal number represents a power of 10, based on its distance from the decimal point. To the left of the decimal point, the powers of 10 increase (10, 100, 1000, ...), while to the right, they decrease (0.1, 0.01, 0.001, ...).

When a number is written in scientific notation, converting it to decimal notation involves moving the decimal point. A positive exponent tells you how many places to move the decimal to the right, as each step increases the number's value by a factor of ten. For instance, the scientific notation for 920 as provided in the exercise is \(9.2 \times 10^2\). To convert this to decimal form, you simply need to move the decimal point two places to the right, resulting in a number without an exponent: 920.

Positive Exponent

A positive exponent, such as the 2 in \(9.2 \times 10^2\), indicates how many times the base number is multiplied by 10. It does not mean to just add zeroes to a number; it means each step is a ten-fold increase in value. Moving the decimal point to the right for a positive exponent makes a number larger because you're increasing its value by powers of ten.

Understanding this concept is vital when dealing with scientific notation. Whenever you see a positive exponent, remember that it serves as an instruction for how many places to shift the decimal point to the right to find the equivalent decimal value. It's an efficient way to write very large or very small numbers in a more compact form.

Base Number

In any expression written in scientific notation, the base number represents the fundamental value, or the 'coefficient,' that is multiplied by a power of 10. The base number is always a number between 1 and 10, which includes one nonzero digit to the left of the decimal point. In the exercise example \(9.2 \times 10^2\), '9.2' is the base number.

The base number in scientific notation is significant because it ensures that scientific notation is standardized. This means when you convert from scientific to decimal notation, you're shifting the precision of the base number, not changing its actual digits. This keeps the integrity of the number consistent while adjusting its scale.