Problem 52 - Beginning and Intermediate Algebra

Question

Convert to scientific notation. $$359,000$$

Step-by-Step Solution

Answer

The short answer is: \(359,000\) in scientific notation is \(3.59 \times 10^5\).

Step 1: Determine the value of \(a\)

To find \(a\), we have to rewrite the given number between \(1\) and \(10\). We achieve this by moving the decimal point, which is currently positioned after the last zero, to the left. Move the decimal point 5 places to the left to obtain \(3.59\). Therefore, \(a = 3.59\).

Step 2: Determine the value of \(b\)

Now we need to find the value of \(b\), which represents an exponential power of 10 that will return the number back to its original form. Since we moved the decimal point 5 places to the left in Step 1, we will have \(10^5\) to move back to the original value. Therefore, \(b = 5\).

Step 3: Write the number in scientific notation

With the values of \(a\) and \(b\) determined, we can now write the original number in scientific notation. Combine the values using the form \(a \times 10^b\): \(3.59 \times 10^5\) The given number, 359,000, in scientific notation is \(3.59 \times 10^5\).

Key Concepts

Understanding the Decimal Point

The decimal point is a tiny dot that plays a big role in mathematical numbers. It's what separates the whole numbers from the fractional part of a number. For example, in the number 3.59, the decimal point comes after the 3. This gives us 3 as the whole number and 59 as the fractional part.

When we convert a number to scientific notation, we often need to move this decimal point. The aim is to place it between the first and second digits, creating a number between 1 and 10. For instance, with 359,000, we shift the decimal point five spots to the left, turning it into 3.59.

This movement of the decimal helps us simplify large numbers, making them easier to work with. The position of the decimal point defines the magnitude of the number.

Exploring Exponentiation

Exponentiation is a concise way of expressing repeated multiplication of a number by itself. When you have a number like 10 multiplied by itself, let's say five times, you can write this as \(10^5\). The number 10 is called the 'base', and 5 is the 'exponent'.

In scientific notation, exponentiation is critical. It's this process that lets us effectively handle very large or very small numbers. When we moved the decimal in 359,000 to write it as 3.59, we needed exponentiation—specifically, multiplying by \(10^5\) to keep the equation balanced.

This means essentially multiplying 3.59 by 10 raised to the power of 5. Through exponentiation, we can expand the number back to its original form, handling numbers with precision and ease.

Understanding Powers of Ten

Powers of ten are a fundamental concept in scientific notation, allowing us to easily scale numbers. When you have a power of ten, it denotes how many times the number 10 is used in multiplication.

For instance, \(10^3\) means 10 multiplied by itself three times, or 1000. In scientific notation, powers of ten are used to express how many times we shifted the decimal point.

In 359,000 expressed as \(3.59 \times 10^5\), the \(10^5\) signifies that if you multiply 3.59 by 100,000 (which is 1 followed by 5 zeros), you would return to the original number. Using powers of ten allows us to write a number in a simplified form, essential for calculations in science and engineering fields.