Question

Write each number in scientific notation. 21,210

Short answer

2.1210 × 10^4

Step-by-step solution

Identify the significant figures

The significant figures in the number 21,210 are 2, 1, 2, and 1.

Place the decimal point

Place the decimal point after the first significant figure to get 2.1210.

Count the number of places the decimal has moved

To turn 21,210 into 2.1210, the decimal has moved 4 places to the left.

Write the number in scientific notation

Combine the figures to form the number in scientific notation: 2.1210 × 10^4.

Key concepts

Significant Figures

Significant figures are the digits in a number that carry meaning and contribute to its precision. When dealing with scientific notation, understanding significant figures is essential. They include:

• All non-zero numbers (e.g., 1, 2, 3)
• Any zeros between significant digits (e.g., 1002 has four significant figures)
• Trailing zeros in a number containing a decimal point (e.g., 50.00 has four significant figures)

For the number 21,210, the significant figures are 2, 1, 2, and 1. Recognizing these helps you accurately convert the number to scientific notation.

Decimal Point Placement

Decimal point placement is a crucial step in forming the correct scientific notation. Once you identify the significant figures, you need to place the decimal point right after the first significant figure.

In our example, 21,210 becomes 2.1210 when the decimal is moved. This adjustment ensures you have a number between 1 and 10, which is essential for scientific notation. So, you take the number 21,210 and transform it to 2.1210, where the decimal point is placed immediately after the first digit.

Exponent Calculation

Exponent calculation denotes how many times you move the decimal point to get the number between 1 and 10. It represents the power of 10 by which you multiply the adjusted significant figures.

In the example of 21,210, moving the decimal point four places to the left changes it to 2.1210. Hence, the exponent here is 4. This movement tells you that your scientific notation will reflect this by using a power of 10:

The alternative number representation becomes 2.1210 multiplied by 10 raised to the power of 4, or \(2.1210 \times 10^{4}\).

Number Representation

Number representation in scientific notation simplifies large or small numbers, making them easier to handle. It is a concise way of expressing numbers as a product of a significant figure and an appropriate power of 10.

In our example, transforming 21,210 into scientific notation requires:

• Identifying the significant figures
• Placing the decimal point
• Calculating the exponent

The final representation will be \(2.1210 \times 10^{4}\). This format makes it straightforward to grasp the scale of the number and its precision, providing a standardized way of communication in mathematical and scientific contexts.