Final answer:
Numbers in scientific notation are expressed with a coefficient and a power of 10 that makes the number easy to read and work with, especially when dealing with very large or very small values. The rules for calculating with scientific notation involve adjusting powers and coefficients accordingly.
Step-by-step explanation:
The task involves performing operations with numbers in scientific notation and expressing the results in that form. Scientific notation is a way of writing numbers that are too big or too small to be conveniently written in decimal form. It is often used by scientists, mathematicians, and engineers.
In normalized scientific notation (called "standard form" in the United Kingdom), the exponent n is chosen so that the absolute value of m remains at least one but less than ten (1 ≤ |m| < 10). Thus 350 is written as 3.5×102. This form allows easy comparison of numbers: numbers with bigger exponents are (due to the normalization) larger than those with smaller exponents, and subtraction of exponents gives an estimate of the number of orders of magnitude separating the numbers. It is also the form that is required when using tables of common logarithms. In normalized notation, the exponent n is negative for a number with absolute value between 0 and 1 (e.g. 0.5 is written as 5×10−1). The 10 and exponent are often omitted when the exponent is 0. For a series of numbers that are to be added or subtracted (or otherwise compared), it can be convenient to use the same value of m for all elements of the series.
Normalized scientific form is the typical form of expression of large numbers in many fields, unless an unnormalized or differently normalized form, such as engineering notation, is desired. Normalized scientific notation is often called exponential notation—although the latter term is more general and also applies when m is not restricted to the range 1 to 10 (as in engineering notation for instance) and to bases other than 10 (for example, 3.15×220)
Here's how you would express the given numbers in scientific notation:
- a. -890,000: Expressed in scientific notation as -8.9 × 10⁵, because -890,000 is -8.9 times 100,000.
- b. 602,000,000,000: Expressed in scientific notation as 6.02 × 10¹², since 602,000,000,000 is 6.02 times 1 trillion, which is 10¹².
- c. 0.0000004099: This is 4.099 × 10⁻⁷ in scientific notation, because 0.0000004099 is 4.099 times one ten-millionth, or 10⁻⁷.
- d. 0.000000000000011: Expressed as 1.1 × 10⁻â in scientific notation, as this number is 1.1 times one hundred trillionth, or 10⁻â.
For sums and differences of numbers in scientific notation, you adjust one of the numbers so both have the same exponent, and then add or subtract the coefficients.