Problem 58 - Elementary Algebra

Question

For the following problems, convert the numbers from scientific notation to standard decimal form. A car traveling at an average speed will travel a distance about equal to the length of the smallest fingernail in \(3.16 \times 10^{-4}\) seconds.

Step-by-Step Solution

Answer

Question: Convert the given number in scientific notation, \(3.16 \times 10^{-4}\) seconds, to standard decimal form. Answer: 0.000316 seconds

Step 1: Identify the given number in scientific notation

The given number in scientific notation is \(3.16 \times 10^{-4}\) seconds.

Step 2: Understand the exponent in the scientific notation

The exponent in the given scientific notation is -4. This means that we need to move the decimal point 4 places to the left to convert it to standard decimal form.

Step 3: Move the decimal point

We move the decimal point 4 places to the left in the number 3.16. In doing so, we add zeros as needed, which gives us the number 0.000316.

Step 4: Write the final answer

The number \(3.16 \times 10^{-4}\) seconds in standard decimal form is 0.000316 seconds.

Key Concepts

Standard Decimal Form

Understanding the standard decimal form is crucial when working with large or small numbers in subjects like science and mathematics. It is the way of writing numbers without using exponents, making them more straightforward to read and use in calculations. When numbers are expressed in standard decimal form, each digit's position corresponds to a specific value based on its place relative to the decimal point. For instance, the digit '5' in the number 503 represents five hundred because it is in the hundreds place.

The process of converting a number from scientific notation to standard decimal form involves expanding the number to show all the places, even if it requires inserting zeros. For example, the scientific notation of a very small length, such as the one mentioned in the exercise - the distance a car travels in a fraction of a second (\(3.16 \times 10^{-4}\) seconds) - appears complex but is a straightforward task to convert to standard decimal form, which would be 0.000316 seconds.

Exponent

An exponent in mathematics is a shorthand way to express how many times a number, called the base, is multiplied by itself. The exponent is usually a small number written slightly above and to the right of the base number. When an exponent is positive, it tells us how many zeroes to add to a number in scientific notation. For instance, the number \(10^3\) means 10 is multiplied by itself three times, which is 1000.

Conversely, a negative exponent, such as -4 in the exercise (\(3.16 \times 10^{-4}\) seconds), tells us we're dealing with a very small number and that we need to move the decimal point to the left to find its standard decimal form. Negative exponents indicate division by the base number raised to the opposite positive power. Therefore, \(10^{-4}\) means 1 divided by 10,000, which would be 0.0001.

Decimal Point Movement

The movement of the decimal point is a pivotal concept when converting between scientific notation and standard decimal form. Moving the decimal point to the left or right changes the value of the number significantly. This operation is closely connected to the concept of exponents: a positive exponent indicates how many places to move the decimal to the right, whereas a negative exponent indicates moving it to the left.

For understanding, take the provided exercise of the speed of a car measured in seconds: \(3.16 \times 10^{-4}\) seconds in scientific notation. The exponent is -4, which tells us to move the decimal point 4 places to the left, resulting in the standard decimal form of 0.000316 seconds. This movement is essential, as it directly translates a number expressed with an exponent into its expanded form, making it easier to work with in regular calculations and comparisons.