(a) The diameter of Earth at the equator is \(7926.381 \mathrm{mi}\). Round this number to three significant figures, and express it in standard exponential notation. (b) The circumference of Earth through the poles is \(40,008 \mathrm{~km}\). Round this number to four significant figures, and express it in standard exponential notation.

Understanding standard exponential notation is crucial in science, especially chemistry, where dealing with very large or small numbers is common. Consider a long number such as the diameter of the Earth, which might have many digits. Representing such numbers succinctly is important both for clarity and accuracy.

Standard exponential notation, also known as scientific notation, is a method of writing numbers as the product of two parts: a coefficient and a power of 10. The coefficient must be a number greater than or equal to 1 and less than 10, and it often includes the significant figures of the original number. The power of 10 indicates how many places the decimal point was moved to get the coefficient.

For example, the Earth's diameter at the equator can be rounded to three significant figures and expressed in standard exponential notation as \(7.93 \times 10^3\) miles. This format is compact and makes it easy to compare different magnitudes of numbers or perform mathematical operations.

When rounding numbers in science, it's essential to consider significant figures, which reflect the precision of a measurement. The process entails looking at the digit immediately to the right of the last significant figure desired. If this digit is five or more, you round up; if it's less than five, you round down.

Rounding 7926.381 miles to three significant figures led to 7930 miles. The '6' after the '2' is the deciding factor; it's greater than five, thus the '2' was rounded up to '3'. This treatment of numbers ensures that when scientists convey measurements, they include only the digits that are reliably known, which helps maintain consistency and accuracy in scientific communication.

Effective rounding in science requires a systematic approach to determine which figures are significant. This approach varies with context, for example, in measurements of length, volume, or mass, and depends on the precision of the instruments used to take these measurements.

In chemistry, working with atoms and molecules often involves very large or very small numbers, making scientific notation indispensable. This system not only simplifies writing and reading such numbers, but also facilitates easier computations during chemical calculations, such as balancing reactions or computing molar masses.

The circumference of the Earth through the poles, when rounded to four significant figures, is \(40,010\) km, or expressed in scientific notation as \(4.001 \times 10^4\) km. This compact form is beneficial in avoiding errors that can occur when dealing with multiple digits. Additionally, scientific notation allows chemists to easily scale quantities in chemical equations, which is especially useful when converting between units, like moles and grams, or when working with Avogadro's number, which is another exceedingly large value commonly used in chemistry.

By using scientific notation, chemists and students alike can focus more on the conceptual understanding of chemical processes rather than getting bogged down by unwieldy numerical values.