Problem 20 - Elementary and Intermediate Algebra

Question

Convert number to standard notation. \(7.63 \times 10^{-5}\)

Step-by-Step Solution

Answer

The standard notation is 0.0000763.

Step 1: Understanding Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is written as the product of a number (between 1 and 10) and a power of ten. Here, the number is 7.63 and the power of ten is \(10^{-5}\).

Step 2: Expanding the Power of Ten

The power of ten, \(10^{-5}\) means that we multiply 7.63 by \(10^{-5}\), which is the same as moving the decimal point five places to the left. This is because a negative exponent indicates division by that power of ten.

Step 3: Moving the Decimal Point

Start with the number 7.63 and move the decimal point five places to the left. As we move left, we add zeros before it to represent the empty place value positions. So 7.63 becomes 0.0000763.

Key Concepts

Understanding Standard Notation

Standard notation is the typical way of writing numbers. It's how we usually see numbers in everyday situations without any special formatting or adjustments for scale. When a number is expressed in scientific notation, such as \(7.63 \times 10^{-5}\), and we convert it back to standard notation, we're essentially returning it to its ordinary decimal representation.
If you've ever heard of numbers like 1, 23, or 456, those are in standard notation. They reflect the value clearly without suggesting any particular power of ten. We use standard notation for better clarity when numbers are neither too large nor too small, making it simple for everyday use.

Converting to Decimal Form

Decimal form is a way to represent numbers without using fractions or scientific notation. It relies purely on base-10 digits placed after a decimal point. To convert a scientific notation number like \(7.63 \times 10^{-5}\) to decimal form, it's crucial to understand the impact of the exponent.
The main idea here is moving the decimal point. Because of the \(10^{-5}\) exponent, we shift the decimal in 7.63 five places to the left. Initially, it is placed immediately after the 7. After moving it five spaces backwards, we need to fill in with zeros to ensure each place value is accounted for, resulting in 0.0000763.
Decimal form simplifies our understanding of the number's scale and makes it easy to compare with other numbers in typical math operations.

Implications of a Negative Exponent

A negative exponent in scientific notation tells us that the number is smaller than one. It shows how many times to divide by ten.
In our exercise, the negative exponent is \(10^{-5}\). This means dividing by \(10^5\) or 100,000. In practical terms, we move the decimal five places to the left from its original position in 7.63. Each move to the left signifies division by ten, which makes the number continually smaller.
The negative exponent is useful for expressing very small quantities that are cumbersome in standard or decimal notation. It's an efficient way to handle these tiny numbers without compromising readability or comprehension.