Problem 76 - Algebra and Trigonometry

Question

\(73-80\) . Write each number in scientific notation. $$ 0.0001213 $$

Step-by-Step Solution

Answer

0.0001213 = 1.213 \times 10^{-4} in scientific notation.

Step 1: Understand Scientific Notation

Scientific notation is a way to express very large or very small numbers. It is written as the product of a number between 1 and 10 and a power of 10. For example, a number can be written as \( a \times 10^n \), where \( 1 \leq a < 10 \) and \( n \) is an integer.

Step 2: Identify the Decimal Placement

For the given number 0.0001213, identify the position to place a single non-zero digit to the left of the decimal. In this case, the first non-zero digit is 1. Thus, we need to move the decimal point after this digit to make it 1.213.

Step 3: Count the Decimal Places Moved

Count the number of places the decimal has been moved to convert from the original number to a number between 1 and 10. For 0.0001213, the decimal is moved 4 places to the right.

Step 4: Write in Scientific Notation

Express the number in scientific notation by noting the new number and power of 10. After moving the decimal 4 places to the right, the number 0.0001213 becomes 1.213 with a power of \(10^{-4}\). Thus, the scientific notation is \(1.213 \times 10^{-4}\).

Key Concepts

Decimal Placement

Understanding decimal placement is essential in scientific notation. The main goal is to pinpoint where to position the decimal so that only one non-zero digit appears to its left. Take the number 0.0001213, for instance. The first non-zero digit is 1, which is crucial. To express this number in scientific notation, we aim to convert it into a number between 1 and 10. We achieve this by shifting the decimal point until it sits right after the first non-zero digit. This transformation results in the number 1.213.

Decimal placement is about reorganizing numbers into a standard form, making them easier to read, comprehend, and compare, especially when dealing with very small or large values. This involves moving the decimal point to create a streamlined version of the original number, favoring a single digit on the left side of the decimal.

Power of 10

The power of 10 is a core part of scientific notation, representing how many times the decimal point is moved. In the case of 0.0001213, once the decimal has been adjusted to form 1.213, we need to express this change in its position using a power of 10.

• If you move the decimal to the left, the power of 10 is positive.
• If you move the decimal to the right, the power of 10 is negative.

For this example, moving the decimal four places to the right results in a power of \[10^{-4}\]. This exponent tells us how many decimal places we moved to establish the scientific notation. Using powers of 10 efficiently compresses the number to a concise form that's manageable and simpler to work with.

Converting Numbers

Converting numbers to scientific notation involves utilizing both the decimal placement and power of 10. It's a straightforward process with significant advantages, especially in dealing with extremes like very large or tiny figures.

Here's how to convert numbers into scientific notation:

• Identify the first non-zero digit and place the decimal after it to form a number between 1 and 10.
• Count the decimal places moved to get from the original number to this new form.
• Use a power of 10 to represent the movement of the decimal point, adjusting for a positive or negative exponent accordingly.

For the number 0.0001213, placing the decimal point yields 1.213, and we then use a power of \(10^{-4}\) to show the shift required. Finally, this number is presented in scientific notation as \(1.213 \times 10^{-4}\). Mastering this skill helps to streamline complex numbers, making them easier to handle and apply in various calculations.