Problem 5 - Algebra for College Students

Question

Write each of the following in scientific notation. For example \(27800=(2.78)(10)^{4}\). \(6,120,000\)

Step-by-Step Solution

Answer

\( 6,120,000 = 6.12 \times 10^6 \)

Step 1: Identify the Significant Figures

Start with the original number: 6,120,000. Identify the significant figures, which are 6.12, excluding the trailing zeros that are just place holders.

Step 2: Determine the Power of Ten

Count how many places the decimal point must move to change the number 6,120,000 into 6.12. You need to move the decimal point 6 places to the left.

Step 3: Write in Scientific Notation

Use the significant figures and the power of ten to express the number in scientific notation. This gives us 6.12 times 10 to the power of 6. So, the scientific notation is: \( 6,120,000 = 6.12 \times 10^6 \)

Key Concepts

Significant Figures

Significant figures are the digits in a number that contribute to its accuracy. When working with large numbers, identifying these figures is crucial. In the number 6,120,000, the significant figures are 6, 1, and 2, which we can express as 6.12. These numbers are significant because they represent the precise digits.
Trailing zeros are not considered significant in whole numbers unless they come after a decimal point. In this number, the zeros serve merely as placeholders for magnitude.
Having a clear understanding of significant figures ensures that your scientific notation is accurate and meaningful.

Power of Ten

The power of ten is a way to express large or small numbers more conveniently as part of scientific notation. It showcases how many times we need to multiply the number 10 to achieve the original size of a number.
In our example, 6,120,000 becomes 6.12, meaning to revert to the original, we need to multiply by a power of ten. Specifically, the power is 6, which tells us the decimal originated from moving six places.
Mathematically, this is denoted as \(10^6\), indicating we multiply by ten, six times.

Decimal Point Movement

Moving the decimal point is essential in converting regular numbers to scientific notation. This movement transforms the number to a standard form that emphasizes significant figures.
To convert 6,120,000 to 6.12, the decimal point shifts six places to the left. Each move to the left decreases the number's magnitude by a factor of ten. We use these shifts to find out the power of ten.
This consistent method ensures a uniform way of converting any large number into scientific notation.

Place Value

Place value is the numerical value that a digit has by virtue of its position in a number. Understanding place value is fundamental when expressing numbers in scientific notation because it helps determine which numbers are significant.
In 6,120,000, the digit 6 is in the millions place, the 1 in the hundred thousands, and the 2 in the ten thousands. These determine the size of the number.
When writing in scientific notation, recognizing the change in place values explains why we need a power of ten to adjust for any movement in the decimal point.