Question

Write this quantity in scientific notation: one tenmillionth of a centimeter

Short answer

Answer: \(10^{-9}\) meters

Step-by-step solution

Determine the numeric value of one tenmillionth of a centimeter

To determine the numeric value of one tenmillionth of a centimeter, we need to convert the wording into a numeric fraction. Ten million is \(10^7\), so one ten millionth is \(\frac{1}{10^7}\) or \(10^{-7}\). Since there are 100 centimeters in a meter, one centimeter equals \(0.01\) meters, or \(10^{-2}\) meters. Therefore, one ten millionth of a centimeter is equal to that fraction multiplied by the size of a centimeter in meters.

Calculate the value of one ten millionth of a centimeter in meters

Multiply the fraction \(10^{-7}\) by the size of a centimeter in meters, which is \(10^{-2}\) meters: \[10^{-7} \times 10^{-2} = 10^{(-7) + (-2)} = 10^{-9}\]

Write the result in scientific notation

The value we calculated in Step 2, \(10^{-9}\), is already in scientific notation, as it is a number between 1 and 10 multiplied by a power of 10. So, one ten millionth of a centimeter is equal to \(10^{-9}\) meters in scientific notation.

Key concepts

Metric Conversions

Understanding metric conversions is crucial for scientific measurements and everyday life. The metric system is a decimal-based system of measurement used around the world. It includes units such as meters for length, grams for mass, and liters for volume, with each unit being 10 times larger or smaller than the next unit. To convert between these units, we use 'powers of ten' which means multiplying or dividing by 10, 100, 1000, and so on. For instance, there are 100 centimeters in a meter, which can be expressed as 1 meter = 100 centimeters or 1 meter = \(10^2\) centimeters. Similarly, to convert between larger and smaller units, like converting kilometers to meters or millimeters to meters, one must multiply or divide by the appropriate power of ten that represents the difference between those units.

Using the exercise as an example, we saw that a centimeter is \(10^{-2}\) meters. This indicates that to convert from centimeters to meters, we need to multiply by the 'power of ten' that corresponds to the centimeter, which in this case, is \(10^{-2}\). Remembering that 'centi' means 'hundredth' can help us recognize why \(10^{-2}\) is used for this conversion.

Powers of Ten

Powers of ten form the basis of the metric system and scientific notation. They are shorthand for writing numbers as a base of ten raised to an exponent. The exponent shows how many times to multiply the base number, ten, by itself. An exponent of 2, written as \(10^2\), means 10 multiplied by 10, which equals 100. Negative exponents indicate division, so \(10^{-2}\) means one divided by 10 twice, which equals 0.01. In metric conversions, powers of ten simplify the process of scaling numbers up or down.

For example, in the given exercise, we multiplied \(10^{-7}\) by \(10^{-2}\) to find the metric value of one ten millionth of a centimeter in meters. When multiplying powers of ten, you simply add the exponents, as seen in \(10^{(-7) + (-2)} = 10^{-9}\). This rule is a fundamental concept in understanding and performing calculations involving powers of ten and is a key element in mastering scientific notation problems.

Scientific Notation Problems

Scientific notation is a convenient way to express large or small numbers. It involves writing numbers as a product of two parts: a coefficient between 1 and 10, and a power of ten. It's especially useful for clearly representing very tiny or very large values, such as atomic measurements or distances in space. When solving scientific notation problems, first identify the coefficient and then determine the appropriate power of ten. There are a few common steps to keep in mind:

• Ensure the coefficient is between 1 and 10.
• Determine the correct power of ten by counting how many places the decimal point needs to move to get the coefficient in this range.
• Multiply these two components to get the number in scientific notation.

In the provided exercise, we found one ten millionth of a centimeter to be \(10^{-9}\) meters. While the number is already in scientific notation, for numbers that aren't, it would be necessary to adjust the coefficient and calculate the power of ten to achieve the correct form. It’s important to practice converting to and from scientific notation to improve problem-solving skills in this area.