Problem 69 - Algebra and Trigonometry

Question

Write each number in decimal notation without the use of exponents. $$ -7.16 \times 10^{6} $$

Step-by-Step Solution

Answer

The decimal notation of \(-7.16 \times 10^{6}\) is \(-7160000\).

Step 1: Identify the exponent

The first step is to identify the exponent, in this case, it's \(6\). This tells you that the decimal point moves 6 places to the right if the exponent is positive, or 6 places to the left if it's negative.

Step 2: Move the decimal point

The second step is to move the decimal point the required number of places. In this case, the exponent is \(6\), so you have to move the decimal point 6 places to the right, resulting in \(-7160000\)

Step 3: Write the number in decimal notation

The final step is to write down the number now that the decimal point has been moved. Make sure not to include the exponent in the final answer.

Key Concepts

Decimal Notation

Decimal notation is the standard way of writing numbers that we are most familiar with. It's how we typically represent numbers using digits from 0-9. Decimal notation can include whole numbers, fractions, or decimals. For example, the number 1234 and the decimal 56.78 are both examples of decimal notation. Unlike scientific notation, decimal notation does not use exponents. Instead, numbers are written fully and clearly for easy readability. When converting from scientific notation to decimal notation, it's crucial to follow the steps to correctly position the decimal point and write the entire number without exponents.

Exponent

An exponent provides information about how many times a number, known as the base, is multiplied by itself. It is written as a small number to the upper right of the base number. In scientific notation, exponents play a key role by showing how many places the decimal point needs to move. For example, in the expression \(10^6\), the number 10 is the base, and 6 is the exponent. This tells us to multiply 10 by itself six times, resulting in 1,000,000.

• An exponent can indicate either a large or very small number, depending on whether it's positive or negative.
• It simplifies expressing large numbers without writing all the zeros.

Decimal Point

The decimal point is a dot used to separate the whole number part from the fractional part of a number. It's an essential symbol in our number system, as it helps indicate values less than one. When converting numbers from scientific notation, the position of the decimal point is crucial. For example, moving the decimal point in a number can entirely change its value, so careful attention is needed when converting. In the expression \(-7.16 \times 10^6\), the decimal point in 7.16 is initially placed after the 7 and before the 1, helping denote its real value in decimal form.

Positive and Negative Exponents

Positive and negative exponents determine the direction and the number of times the decimal point moves. A positive exponent moves the decimal point to the right, making the number larger. A negative exponent shifts the decimal point to the left, resulting in a smaller number.

• Positive exponents mean you multiply; for example, \(10^3\) means multiply 1 by 10 three times (1000).
• Negative exponents signify division; for example, \(10^{-2}\) means divide 1 by 10 squared, resulting in 0.01.

Understanding these roles of exponents helps in moving between scientific and decimal notation easily and accurately. In our exercise, a positive exponent of 6 moved the decimal in \(-7.16\) to the right, resulting in \(-7,160,000\).