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Chapter 10: Problem 603

Write each number in scientific notation. $$0.00814$$

Short Answer

Expert verified
8.14 \times 10^{-3}

Step by step solution

01

- Understanding Scientific Notation

Scientific notation expresses a number as a product of a coefficient (between 1 and 10) and a power of 10. The general form is: \[ a \times 10^n \] where \(1 \leq |a| < 10\) and \(n\) is an integer.
02

- Move the Decimal Point

To convert \(0.00814\) to scientific notation, move the decimal point to the right until it is positioned after the first non-zero digit. In this case, moving the decimal 3 places to the right gives \(8.14\).
03

- Adjust the Power of 10

Each movement of the decimal point to the right is represented as a negative power of 10. Since the decimal was moved 3 places, the power of 10 will be \(-3\).
04

- Write in Scientific Notation

Combine the coefficient and the power of 10: \[ 0.00814 = 8.14 \times 10^{-3} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

decimal point
The decimal point is a crucial part of understanding scientific notation. It separates the whole number part from the fractional part in a decimal number. For example, in the number 0.00814, the decimal point is located to the left of all non-zero digits.

When converting to scientific notation, the goal is to move the decimal point so that only one non-zero digit is to its left. This helps create a coefficient that falls between 1 and 10. For instance, moving the decimal point three places to the right in 0.00814 gives us 8.14, which is in the proper range for a coefficient.
power of 10
In scientific notation, the power of 10 indicates how many places the decimal point was moved to create the coefficient. This can either be a positive or negative integer.

Moving the decimal point to the right decreases the exponent (making it more negative), while moving it to the left increases the exponent. Each shift of the decimal point by one place represents a change in the power of 10 by one.

For example, in the number 0.00814, moving the decimal point three places to the right to get 8.14 means the power of 10 should be -3. Therefore, the scientific notation for 0.00814 is written as \(8.14 \times 10^{-3}\).
coefficient
The coefficient in scientific notation is the number that falls between 1 and 10 (inclusive of 1 and less than 10). It represents the significant digits of the original number.

For the number 0.00814, the coefficient is obtained by repositioning the decimal point to directly follow the first non-zero digit. After moving the decimal point three places to the right, our coefficient becomes 8.14.

Thus, the number 0.00814 is represented in scientific notation as \(8.14 \times 10^{-3}\), where 8.14 is the coefficient, and -3 is the power of 10.

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