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. \(4 \times 10^5\) b. \(3.9 \times 10^5\) c. \(3.86 \times 10^5\) d. \(3.8550 \times 10^5\)

Step-by-step solution

Determine Exponential Notation for 385,500

To express 385,500 in exponential notation, we first need to rewrite it as a product of a coefficient and a power of 10. The coefficient will be a decimal number between 1 and 10. The exponent indicates how many places the decimal point should be moved. In this case, the decimal point will be moved 5 places to the left to get a number between 1 and 10 (3.855). So, the exponential notation for 385,500 is \(3.855 \times 10^5\).

(a) One Significant Figure

To express 385,500 in exponential notation with one significant figure, we need to round the coefficient to just one digit. The coefficient is 3.855, and rounding it to 1 digit gives 4 (since 0.855 rounds up to 1). Therefore, the number 385,500 expressed in exponential notation with one significant figure is \(4 \times 10^5\).

(b) Two Significant Figures

To express 385,500 in exponential notation with two significant figures, we need to round the coefficient to two digits. The coefficient is 3.855, and rounding it to 2 digits gives 3.9. Therefore, the number 385,500 expressed in exponential notation with two significant figures is \(3.9 \times 10^5\).

(c) Three Significant Figures

To express 385,500 in exponential notation with three significant figures, we need to round the coefficient to three digits. The coefficient is 3.855, and rounding it to 3 digits gives 3.86 (since 0.005 rounds up to 0.01). Therefore, the number 385,500 expressed in exponential notation with three significant figures is \(3.86 \times 10^5\).

(d) Five Significant Figures

To express 385,500 in exponential notation with five significant figures, we need to include trailing zeros to the coefficient (since there are no more nonzero digits). The coefficient is 3.855, and adding two trailing zeros gives 3.8550. Therefore, the number 385,500 expressed in exponential notation with five significant figures is \(3.8550 \times 10^5\).