Problem 6 - Algebra for College Students

Question

Write each of the following in scientific notation. For example \(27800=(2.78)(10)^{4}\). \(72,400,000\)

Step-by-Step Solution

Answer

\(7.24 \times 10^7\)

Step 1: Identify the Non-Zero Digits

Identify the significant digits in the number. Here, the number is 72,400,000, which has significant digits 7, 2, and 4.

Step 2: Place the Decimal Point

Insert a decimal point after the first non-zero digit to rewrite the number as a decimal. Place the decimal after 7 to get 7.24.

Step 3: Count the Number of Decimal Places

Count how many places you move the decimal point to form the number from its original position. Moving from 72,400,000 to 7.24 involves 7 places to the left.

Step 4: Write Using Scientific Notation

Express the number in scientific notation as the coefficient and a power of ten. Here, it's written as \(7.24 \times 10^7\) because the decimal moved 7 places to the left.

Key Concepts

Significant Digits

Significant digits, or significant figures, represent the precision of a number. They include all the non-zero numbers as well as any zeros that appear between them or after the decimal point in a measurement. In scientific notation, it's crucial to correctly identify these digits to maintain a number's accuracy.

• In the number 72,400,000, the significant digits are 7, 2, and 4.
• These digits reflect the specific value without including any leading or trailing zero placeholders.

Understanding significant digits ensures we accurately convey quantities without introducing false precision.

Decimal Point Placement

Decimal point placement is a vital step in converting a standard number to scientific notation. You need to adjust the decimal point so that only one non-zero digit remains to the left of the point.

• For our example, 72,400,000 becomes 7.24 when the decimal point is correctly placed.
• This placement allows us to represent the number in a way that highlights its significant figures and simplifies larger numbers.

Proper decimal point placement enhances clarity and consistency when conveying large or small numbers.

Power of Ten

The power of ten in scientific notation reflects how many places the decimal point has moved to convert the number. It indicates the scale of the original number, helping to simplify it while maintaining accuracy.

• In 72,400,000, the decimal moves 7 places to the left, resulting in a power of ten of 7.
• The notation becomes \(10^7\), indicating a large-scale move to the left, transforming the large number to a compressed form.

Using powers of ten, we can easily handle extraordinarily large or small numbers, making comparisons and calculations more manageable.

Exponential Form

Exponential form is the method by which we express numbers in a compact and consistent manner using scientific notation. It typically involves a coefficient (the significant digits) and a power of ten that denotes the scale.

• For the number 72,400,000, the scientific notation is \(7.24 \times 10^7\).
• This form enables quick understanding of the number's size and precision without overwhelming digits.

Exponential form is invaluable in scientific fields for conveying large data or small measurements clearly and efficiently.