A rock is experimentally determined to be \(3.5\) billion years old, with an uncertainty of plus on minus 100,000 years. Write the age of this rock in scientific notation.

Scientific notation is a method used by scientists and mathematicians to express very large or very small numbers in a compact, precise form. It is especially useful in fields like chemistry, where measurements can span a broad range of values. The basic format of scientific notation is to take a number and place it between 1 and 10, multiplied by a power of 10.
For instance, let's consider the number 6,500, which in scientific notation is written as \(6.5 \times 10^3\). This indicates that you'd multiply 6.5 by 10, three times (which is 10 cubed), to get back to the original number. The exponent, in this case, 3, indicates how many places the decimal point has been moved to the left for large numbers or to the right for small numbers.
Scientific notation is not just for simplification but also to assure accuracy and precision in scientific calculations. When converting large figures or very minute measures into scientific notation, it's crucial to keep significant figures in mind, as they convey how precise a measurement is.

Measurement uncertainty refers to the doubt that exists whenever a measurement is made in science. It acknowledges that no measurement is perfectly precise and quantifies the potential variation in measurements. For instance, if a chemist measures a substance and finds it to be 7 grams, the true weight could be slightly more or less, due to instruments accuracy, or human error. This could be reported as \(7 \pm 0.1\) grams, where 0.1 grams is the uncertainty.
In scientific notation, the uncertainty is typically expressed with the same exponent as the main measurement. Taking the age of the rock from our earlier example, the uncertainty was 100,000 years, which would be in scientific notation as \(1 \times 10^5\) years, where the exponent matches that of the primary value when expressed in the same terms of significant figures.
Understanding and reporting uncertainty is vital in scientific communication. It allows others to gauge the reliability of the measurements and compare them to other data accurately.

Expressing the age of objects, such as a rock, in scientific notation allows for a clear and concise representation, particularly when dealing with time spans of millions or billions of years. In our example, the age of the rock is 3.5 billion years, which appears as \(3.5 \times 10^9\) years in scientific notation. The exponent 9 signifies the number of times you would move the decimal point to the right to convert it back to standard notation.
It's also important to express the uncertainty associated with the measurement in the same format. So, for an uncertainty of 100,000 years, we use \(1 \times 10^5\) years to maintain consistency and precision. By combining these, we indicate the age of the rock, including its uncertainty, as \(3.5 \times 10^9 \pm 1 \times 10^5\) years. This precise method of expressing age communicates not only the magnitude of the time involved but also the potential variance clearly and efficiently, which is essential in geological and other scientific research where exact ages are crucial.