Problem 31 - Elementary and Intermediate Algebra

Question

Write number in scientific notation. \(1,700,000\)

Step-by-Step Solution

Answer

The number 1,700,000 in scientific notation is \(1.7 \times 10^6\).

Step 1: Understand Scientific Notation

Scientific notation is a way of expressing really large or really small numbers as the product of a number between 1 and 10 and a power of 10.

Step 2: Identify the Significant Figures

The number 1,700,000 has the significant figures 1.7, since it can be expressed as 1.70 multiplied by some power of 10 to represent where the decimal is placed.

Step 3: Determine the Power of 10

Count how many places you move the decimal point from its original position in 1,700,000, where it is at the end of the number, to right after the first non-zero digit (1). The decimal moves 6 places to the left, so the exponent is 6.

Step 4: Write the Scientific Notation

Combine the significant figures and the power of 10 to write the number in scientific notation: the number 1.7 multiplied by 10 raised to the power of 6. This is written as: \[1.7 \times 10^6\]

Key Concepts

Understanding Significant Figures

Significant figures are the digits in a number that contribute to its precision. When we take a number like 1,700,000, we want to express only the most crucial digits that show the value accurately, especially when converting to scientific notation. In this case, the significant figures are 1.7.

The idea is simple: focus on digits that matter. Any zeros in the number, which don't affect the overall value but just change the scale, aren't considered significant. For example:

• In 1,700,000, the digits 1 and 7 are significant.
• The trailing zeros help position the decimal but aren't considered significant here.

This helps in scientific computations where precision and accuracy are important because it reduces unnecessary complexity.

The Concept of Power of 10

The power of 10 is a fundamental concept in expressing large numbers succinctly. It denotes how many times the number 10 is being multiplied by itself. In scientific notation, moving the decimal point changes the power of 10.

Here's how it works with our number, 1,700,000:

• The decimal point starts at the end of the number: 1,700,000.
• To express this in a simpler form, move the decimal 6 places to the left so it sits after the 1: 1.7.
• Every step left means increasing a power of 10. Hence, we write \(10^6\).

By using the power of 10, we can compress large numbers into more manageable formats, such as turning 1,700,000 into \(1.7 \times 10^6\). It makes calculations simpler and reading numbers much easier.

Expressing Large Numbers with Scientific Notation

Expressing large numbers in scientific notation is a common practice in math and science. It simplifies the way we handle big numbers by reducing them to a format that's easier to read and work with.

Let's see how scientific notation works:

• Start with a large number, like 1,700,000.
• Identify its significant figures, which are 1.7.
• Determine how far you moved the decimal point to reach these figures (in this case, 6 places to the left).
• Use that movement to find the power of 10 (\(10^6\)).

By combining the significant figures with the power of 10, scientific notation allows us to write 1,700,000 as \(1.7 \times 10^6\).
This system is not only more concise but also maintains the number's accuracy and makes it easier to manage calculations, preserving both value and comprehensibility.