Problem 73 - Algebra A Combined Function

Question

Write each number in scientific notation. $$ 78,000 $$

Step-by-Step Solution

Answer

78,000 in scientific notation is \(7.8 \times 10^4\).

Step 1: Identifying the Significant Figures

Begin by identifying the significant figures in the number 78,000. The significant figures here are 78, as these are non-zero digits.

Step 2: Placing the Decimal Point

Place a decimal point after the first significant digit to convert the number into a form that is suitable for scientific notation. The number becomes 7.8.

Step 3: Counting the Powers of Ten

Count the number of digits that follow the first significant digit in the original number. There are four digits (80,000) following the '7.' This means moving the decimal four places to get from 7.8 to 78,000.

Step 4: Writing in Scientific Notation

Combine the adjusted significant figure with the power of ten calculated. The number in scientific notation is written as \(7.8 \times 10^4\).

Key Concepts

Understanding Significant Figures

Significant figures are essential in expressing a number in scientific notation accurately. They include all non-zero digits and any zeros that are trapped between them. In the number 78,000, the significant figures are '78'. Here, the zeros are merely placeholders. Identifying significant figures is crucial as it dictates the digits that you'll eventually express in scientific notation.

• Non-zero digits are always significant.
• Any zeros between significant figures are significant.
• Leading zeros are not significant, and trailing zeros might only be significant if there's a decimal point explicitly showing precision.

By recognizing the significant figures, you can ensure precision and accuracy in the number's scientific representation.

The Role of Powers of Ten

The power of ten in scientific notation reflects how many places the decimal point needs to move to express the number in a standard form. In the example of 78,000, after identifying the significant figures as 7.8, you count how many places the digits had originally to move back from 7.8 to become 78,000. There are four places, so the power of ten is 4. Therefore, the number 78,000 becomes \(7.8 \times 10^4\) in scientific notation. This technique of using powers of ten allows us to express both very large and very small numbers in a concise form. In scientific notation:

• Positive powers of ten indicate large numbers.
• Negative powers of ten represent small numbers.
• The exponent tells you how many times the decimal point has been moved.

Decimal Point Placement in Scientific Notation

The decimal point placement is fundamental when converting a standard number into scientific notation. The position of the decimal point should be after the first significant figure, which ensures precision. In our example, when converting 78,000, the decimal point shifts from after 78 (in its full form) to 7.8. This action of placing the decimal point is what helps to simplify the number into scientific notation. Here's how to do it:

• After identifying the significant figures, place the decimal point directly after the first significant digit.
• Count how many places the decimal has moved from its new position back to the original number.
• This count becomes the power of ten in the scientific notation.

Mastering decimal point placement not only facilitates precise scientific notation but also reinforces an understanding of the number's true value.