Problem 23 - College Algebra with Corequisite Support

Question

For the following exercises, convert each number in scientific notation to standard notation. \(1.6 \times 10^{10}\)

Step-by-Step Solution

Answer

The standard notation of \(1.6 \times 10^{10}\) is 16,000,000,000.

Step 1: Understand Scientific Notation

Scientific notation is a way to express very large or very small numbers. It is written as the product of a number (usually between 1 and 10) and a power of 10. In this exercise, the number is expressed as \(1.6 \times 10^{10}\).

Step 2: Identify the Components

In the expression \(1.6 \times 10^{10}\), we have two components: the coefficient \(1.6\) and the exponent \(10\). The exponent indicates how many places the decimal point will move.

Step 3: Apply the Exponent to the Coefficient

Since the exponent is \(10\), move the decimal point in \(1.6\) 10 places to the right. As you move, add zeros to fill in the gaps as needed.

Step 4: Convert to Standard Notation

Move the decimal point 10 places to the right: 1.6 becomes 16000000000 by adding nine zeros. Therefore, the standard notation is 16,000,000,000.

Key Concepts

Standard Notation Explained

Standard notation is what we commonly use to express numbers in everyday life, such as the number 546 or 23,000. Unlike scientific notation, it doesn't involve exponents. Instead, it's the number written out in full with all its digits. For example, if you were to convert scientific notation into standard notation, you would write out the number completely with its zeros and decimal point in the correct positions. By learning to convert from scientific to standard notation, you gain an appreciation for how immensely large or tiny numbers can be represented simply and accurately.

Understanding Exponents

In mathematics, an exponent indicates how many times a number, known as the base, is multiplied by itself. For example, in the expression \(10^{3}\), 10 is the base and 3 is the exponent, meaning you multiply 10 by itself three times: \(10 \times 10 \times 10 = 1000\).
Exponents are essential in scientific notation as they show how many decimal places to move. In \(1.6 \times 10^{10}\), the exponent is 10, which tells you to move the decimal 10 places to the right when converting to standard notation.
This movement of the decimal either expands the number into something large or contracts it to something very small, all based on the positive or negative value of the exponent.

The Decimal Point's Role

The decimal point in a number helps separate the whole number part from the fractional part. When working with scientific notation, it indicates where the initial number starts. For instance, in \(1.6 \times 10^{10}\), the decimal point shows that our starting point is 1.6.
Converting from scientific to standard notation involves moving the decimal point across the number based on the exponent. If the exponent is positive, move the decimal to the right, creating a larger number. If it's negative, you move it to the left to create a small number.

• Positive exponent: move right.
• Negative exponent: move left.

This understanding helps you know exactly where to place extra zeros or adjust the digits to get the final standard number.

Defining Coefficients

Coefficients are numbers multiplied by a power of ten in scientific notation. In \(1.6 \times 10^{10}\), 1.6 is the coefficient. It can be any number between 1 and 10, including decimals. The coefficient is crucial because it represents the significant digits of the original number.
Multiplying the coefficient by \(10^{x}\) effectively scales the number up or down according to the exponent. It's like a weight or a measure of how much we should expand or contract the number in question as we convert to or from scientific notation.