Convert the following numbers into scientific notation: a. \(93,000,000\) b. 708,010 c. \(0.000248\) d. \(800.0\)

Exponents are an essential component of scientific notation. They represent the number of times that a base (in this case, 10) is multiplied by itself. For instance, when we say \(10^3\), it means that 10 is multiplied by itself three times: \(10 \times 10 \times 10\).

In scientific notation, exponents allow us to express very large or very small numbers compactly. A positive exponent shows that the decimal point has moved to the left in large numbers, indicating multiplication by a power of ten. Contrarily, a negative exponent, like in the case of \(2.48 \times 10^{-4}\), suggests the decimal point has moved to the right in small numbers, which is the same as dividing by a power of ten.

Exponents are not just placeholders. They give us valuable information about the scale of a number. For instance, in the number \(9.3 \times 10^7\), the exponent 7 signifies that the decimal point has been shifted seven places to the left, indicating a much larger value than the initial 9.3.

Understanding decimal place movement is crucial when converting numbers to scientific notation. This process involves positioning the decimal point so that there's only one non-zero digit to the left of it. For example, to convert the number 93,000,000 into scientific notation, we move the decimal place seven spaces to the left to get 9.3.

When dealing with numbers less than 1, such as 0.000248, the decimal point moves to the right instead. Here, the decimal moves four places to the right to place it after the 2, yielding 2.48. It's important to track the number of places the decimal moves since this directly determines the exponent in scientific notation.

Quick Tip:

For numbers greater than 1, the exponent will be positive and equal to the number of places moved. If the number is less than 1, the exponent will be negative. This method simplifies the way we write and work with numbers that have many zeroes, which can be easy to miscount or misinterpret.

Standard form is another term for scientific notation, and it refers to the systematic way of writing numbers that are too large or too small to be conveniently expressed in decimal form. Scientific notation adheres to a standard: it expresses numbers as a product of two factors - a coefficient and a power of 10.

The coefficient must be a number between 1 and 10, including 1 but excluding 10, and is followed by a multiplication sign and a power of 10, which simplifies the complex original number. For instance, 708,010 is written in standard form as \(7.08010 \times 10^5\).

This practice is not just an academic exercise; it has practical applications in various scientific fields. Scientists and engineers frequently encounter extremely precise measurements or astronomical distances that are far more manageable when expressed in standard form. By reducing these numbers to a base coefficient and an exponent, computations and comparisons become much more straightforward.