Problem 9 - Elementary Algebra

Question

Write the following numbers in scientific notation. $$ 0.000098001 $$

Step-by-Step Solution

Answer

Question: Convert the number 0.000098001 into scientific notation. Answer: The scientific notation of the given number is 9.8001 × 10^(-4).

Step 1: Identify the number to be converted

The given number is 0.000098001.

Step 2: Determine the exponent

To put the number in scientific notation, we need to move the decimal point to the right until there's only one non-zero digit to its left. Count how many places you move the decimal point; this will be the exponent. In this case, we move the decimal 4 places to the right. Since we moved to the right, the exponent will be negative. Therefore, the exponent is -4.

Step 3: Write the number in scientific notation

Now that we know the exponent, we can write the given number in scientific notation. To do this, we rewrite the number with the decimal point moved and then multiply it by 10 raised to the negative exponent we found. Combining our findings, the scientific notation of the given number is: $$ 9.8001 \times 10^{-4} $$

Key Concepts

Understanding the Decimal Point

The decimal point plays a crucial role in expressing numbers in scientific notation. It is the dot that separates the whole number part from the fractional part of a number. When converting a number into scientific notation, it is important to manipulate the position of the decimal point correctly.
For example, with the number 0.000098001, we move the decimal point to create a number where only one non-zero digit appears to its left. For this, we shift the decimal point 4 places to the right. Moving the decimal point transforms the number, but it does not change its value when we represent it in scientific notation.
In scientific notation, the objective is to have a single non-zero digit to the left of the decimal point, like in our example where the number becomes 9.8001. The decimal point's position determines the value of the exponent in a scientific notation expression.

The Role of Exponents

Exponents are key in scientific notation because they tell us how many times the number has been multiplied or divided by 10. After moving the decimal point in a number, the exponent specifies how many places the decimal point has been moved and in which direction. In this context, it's an integral part of expressing large or small numbers concisely.
Returning to our example, after moving the decimal point 4 places to the right, we express this shift with an exponent of \(-4\). This indicates that the decimal point was moved four spaces to the right, emphasizing the initial smallness of 0.000098001. When you see a number in scientific notation, such as \(9.8001 \times 10^{-4}\), the exponent \(-4\) functions as an indicator of that adjustment.

Explaining Negative Exponents

Understanding negative exponents is essential when dealing with scientific notation. A negative exponent indicates the inverse of multiplication—that is, division. This plays a role when the decimal point in a number moves to the right in the conversion process, signifying the original number was less than one.
In our specific example, shifting the decimal point 4 places to the right to convert 0.000098001 into scientific notation changes the number to \(9.8001 \times 10^{-4}\). The \(-4\) is a negative exponent, showing that the decimal shift required division by 10 four times.
This exponent is essential for accurately reflecting the size of the original number in its new form. By understanding negative exponents, you better grasp how scientific notation functions for tiny numerals.