What is the product of $\left(7.9 \times 10^{11}\right)\left(3.5 \times 10^6\right)$ written in scientific notation?
A. $11.4 \times 10^{17}$
B. $2.765 \times 10^{19}$
C. $2.765 \times 10^{17}$
D. $2.765 \times 10^{18}$
Answer (1)
Multiply the decimal parts: 7.9 × 3.5 = 27.65 .
Multiply the powers of 10: 1 0 11 × 1 0 6 = 1 0 17 .
Combine the results: 27.65 × 1 0 17 .
Adjust to scientific notation: 2.765 × 1 0 18 .
The product in scientific notation is 2.765 × 1 0 18 .
Explanation
Problem Analysis We are asked to find the product of two numbers expressed in scientific notation and express the result in scientific notation as well. The given numbers are 7.9 × 1 0 11 and 3.5 × 1 0 6 .
Multiplication Strategy To find the product, we multiply the decimal parts and the powers of 10 separately. That is, we compute ( 7.9 × 3.5 ) × ( 1 0 11 × 1 0 6 ) .
Multiplying Decimal Parts First, let's multiply the decimal parts: 7.9 × 3.5 = 27.65 .
Multiplying Powers of 10 Next, let's multiply the powers of 10: 1 0 11 × 1 0 6 = 1 0 11 + 6 = 1 0 17 .
Combining Results Now, we multiply the results from the previous two steps: 27.65 × 1 0 17 .
Adjusting to Scientific Notation To express this in scientific notation, the decimal part must be between 1 and 10. Since 27.65 is greater than 10, we need to adjust the decimal point. We can rewrite 27.65 as 2.765 × 1 0 1 . Therefore, 27.65 × 1 0 17 = ( 2.765 × 1 0 1 ) × 1 0 17 = 2.765 × 1 0 1 + 17 = 2.765 × 1 0 18 .
Final Answer Thus, the product of ( 7.9 × 1 0 11 ) ( 3.5 × 1 0 6 ) in scientific notation is 2.765 × 1 0 18 .
Examples
Scientific notation is extremely useful in fields like astronomy and physics, where dealing with very large or very small numbers is common. For example, the distance to the nearest star, Proxima Centauri, is approximately 4.017 × 1 0 13 kilometers. The mass of an electron is approximately 9.109 × 1 0 − 31 kilograms. Using scientific notation makes these numbers easier to write, read, and perform calculations with.
