Question

Use exponential notation to express the number 385,500 to a. one significant figure. b. two significant figures. c. three significant figures. d. five significant figures.

Short answer

a. The number 385,500 expressed to one significant figure in exponential notation is \(4 \times 10^5\). b. The number 385,500 expressed to two significant figures in exponential notation is \(3.9 \times 10^5\). c. The number 385,500 expressed to three significant figures in exponential notation is \(3.86 \times 10^5\). d. The number 385,500 expressed to five significant figures in exponential notation is \(3.8550 \times 10^5\).

Step-by-step solution

Find the largest power of 10

For one significant figure, the largest power of 10 is \(10^5\), because \(385,500 \approx 4 \times 10^5\).

Round the number

Round 385,500 to the nearest multiple of \(10^5\). The closest multiple is 400,000.

Write in exponential notation

Express 400,000 in exponential notation: \[4 \times 10^5\] #b. two significant figures#

Find the largest power of 10

For two significant figures, the largest power of 10 is \(10^4\), because \(385,500 \approx 39 \times 10^4\).

Round the number

Round 385,500 to the nearest multiple of \(10^4\). The closest multiple is 390,000.

Write in exponential notation

Express 390,000 in exponential notation: \[3.9 \times 10^5\] #c. three significant figures#

Find the largest power of 10

For three significant figures, the largest power of 10 is \(10^3\), because \(385,500 \approx 386 \times 10^3\).

Round the number

Round 385,500 to the nearest multiple of \(10^3\). The closest multiple is 386,000.

Write in exponential notation

Express 386,000 in exponential notation: \[3.86 \times 10^5\] #d. five significant figures#

Count the number of significant figures

The given number, 385,500, has five significant figures.

Write in exponential notation

Since the given number already has five significant figures, write it in exponential notation: \[3.8550 \times 10^5\]

Key concepts

Significant Figures

When we talk about significant figures, we're referring to the digits in a number that carry meaning or contribute to its precision. These figures include all non-zero numbers, any zeros between significant digits, and any trailing zeros in the decimal portion. The significance is crucial in scientific and engineering calculations because it indicates the accuracy of measurements. For example, in the number 385,500, when reduced to one significant figure, it becomes 4 (because the number is closest to 4 times a large power of 10). But with two figures, it becomes 3.9 because it's nearest to 39 times a smaller power of ten. Understanding how many significant figures to use helps ensure that results are both precise and meaningful.

Scientific Notation

Scientific notation is a method to express very large or very small numbers conveniently. In this format, numbers are written as a product of a number between 1 and 10 and a power of ten. This notation is especially useful in sciences where such scales are common. For example, 385,500 can be represented as \(3.855 \times 10^5\). It makes the digit count more manageable and simplifies both calculations and comparisons. By reducing or expanding the number of significant digits, this format helps in maintaining the necessary level of precision without cumbersome zeros.

Rounding Numbers

Rounding numbers simplifies them while maintaining proximity to the actual value. The rounding depends heavily on the context, such as reports or scientific analysis. For 385,500, if limiting to the nearest thousand, it rounds to 386,000, which in turn is written as \(3.86 \times 10^5\) in scientific notation. Rounding can greatly affect the results of computations, especially when working with more than one rounded figure. The concept of significant figures often helps decide how much rounding is appropriate without losing essential information.

Powers of Ten

Powers of ten refer to numbers expressed using a base of ten and an exponent, indicating how many times the base is multiplied by itself. Using powers of ten is integral to exponential notation and scientific notation. The number 385,500 can be expressed using powers of ten as \(3.855 \times 10^5\). This is useful as it provides a clear and concise way to display large numbers and understand their scale. It's much easier to work mathematically with numbers in this form, making computations more efficient and less error-prone.