Problem 13 - Algebra 1: Concepts and Skills

Question

Write the number in scientific notation. $$ 0.0205 $$

Step-by-Step Solution

Answer

The decimal number 0.0205 in scientific notation is \(2.05 \times 10^{-2}\).

Step 1: Identify the digit

Locate the first non-zero digit in the decimal. In this case, the first non-zero digit in 0.0205 is 2.

Step 2: Write the digit

Write the number that starts with that digit. Exclude any leading zeros. Here it means to write 2.05.

Step 3: Determine the power of 10

Figure out the power of 10. Since we have moved the decimal point two places to the right, we use a negative power of 10. Hence, the power of 10 is -2.

Step 4: Write in scientific notation

Combine the number from step 2 with the power of 10 from step 3 to express the number in scientific notation. Hence, the number 0.0205 in scientific notation is \(2.05 \times 10^{-2}\).

Key Concepts

Decimal Representation

A decimal representation is a way of expressing numbers that are not whole numbers. When you see a number like 0.0205, it's displayed in decimal form. This format utilizes a decimal point to indicate fractions or portions of a base-10 number system. The digits following the decimal point represent parts of a whole number, each holding a specific value depending on its position.

For example, in the number 0.0205, the digit '2' is in the hundredths place, meaning it represents 2 parts of 100. The digit '0' following it is in the thousandths place, and the '5' is in the ten-thousandths place, representing 5 parts of 10,000.

Understanding decimal representation helps to visualize small quantities and is essential in converting them into scientific notation.

Power of 10

The power of 10 is a crucial part of scientific notation. It reflects how many places you need to move the decimal to express the number in standard or scientific form.

In our example, the number 0.0205 is written as 2.05 in scientific notation. To arrive at this, we move the decimal two places to the right. This movement finds us a new form of the number, which is easier to work with in various calculations.

Each move of the decimal point corresponds to a power of 10, which is why we say the number is multiplied by 10 raised to a specific power. This makes the number more compact and provides a clearer sense of its order of magnitude.

Negative Exponent

In scientific notation, a negative exponent indicates the direction and number of places the decimal was moved to convert the number. It tells us how many times we should divide the number by 10 to get back to its original value.

For the number 0.0205, when we express it as 2.05, we note that the decimal was moved two places to the right. Therefore, the power of 10 is -2.

• Negative emphasizes moving the decimal to the right.
• It suggests a small fraction.

By using negative exponents, we can compactly write very tiny numbers like 0.0205 as \(2.05 \times 10^{-2}\), helping to simplify expressions and calculations. Negative exponents are particularly useful in expressing very small numbers in a manageable form.