Problem 63 - Prealgebra

Question

Write each number in standard form. $$3.05 \times 10^{6}$$

Step-by-Step Solution

Answer

The number is 3,050,000 in standard form.

Step 1: Understanding Scientific Notation

The number is given in scientific notation, which is used to express very large or very small numbers. The number is currently written as \(3.05 \times 10^{6}\). Here, \(3.05\) is the coefficient, and \(10^{6}\) is the power of 10.

Step 2: Applying the Power of 10

To convert the number from scientific notation to standard form, we need to multiply \(3.05\) by \(10^{6}\). This means moving the decimal point in \(3.05\) six places to the right, since the exponent is positive.

Step 3: Moving the Decimal Point

Start with the number 3.05 and move the decimal point six places to the right: 1. From 3.05 to 30.5 (1st place) 2. 305 (2nd place) 3. 3050 (3rd place) 4. 30500 (4th place) 5. 305000 (5th place) 6. 3050000 (6th place) Once you move the decimal point six positions, the number becomes 3,050,000.

Step 4: Ensuring Correct Standard Form

Make sure that after moving the decimal point, no digits are left out. The resulting number, 3,050,000, is the standard form, meaning it is a regular whole number without any decimal places.

Key Concepts

Standard Form

Standard form is a straightforward way to write numbers. It involves expressing numbers without exponents, right as you see them in everyday life. In the context of this exercise, the number needs to be expressed entirely using digits, like 3,050,000, instead of scientific notation.
Scientific notation can be convenient for displaying very large or small numbers, as it breaks them down into a manageable format. However, standard form makes the number easier to read and understand at a glance.

• In scientific notation, a number is written as a coefficient and a power of 10, while in standard form, it's written in full.
• Standard form reflects how numbers are typically displayed in daily documents, textbooks, and reports.
• Numbers are displayed without abbreviations or mathematical terms, which makes communication clearer.

Power of 10

The power of 10 is an integral part of scientific notation. It tells you how many times to multiply the coefficient by 10.
In our example, the power provided is 6, as shown in the form of the exponent in the term \(10^{6}\).

• A positive exponent indicates how many times to multiply the coefficient by 10, effectively moving the decimal point to the right by that number of places.
• As the power of 10 increases, the number represented grows significantly larger. Conversely, a negative exponent would shrink it.
• In this problem, applying \(10^{6}\) means transforming the number from a simple decimal to a large whole number (i.e., 3,050,000).

Understanding the power of 10 aids in quickly converting numbers from one form to another. This concept is powerful in fields like astronomy and physics where large numbers are common.

Decimal Point Placement

Moving the decimal point is a crucial step in converting scientific notation to standard form. The placement of the decimal determines the actual size of the number.
Here, you're instructed to move the decimal point 6 places to the right, starting from 3.05, as the power of 10 indicated is 6.
This process looks like:

• Begin at 3.05.
• Shift the decimal after every digit until it moves 6 spaces.

Once the decimal point is where it belongs, add zeros if necessary to fill any spaces, resulting in the number 3,050,000.
Including the correct amount of zeros reveals the true magnitude of the number once it exits scientific notation. This ensures the number is accurately placed on the number scale, crucial when precise values are needed.