Find the sum of $3.2 \times 10^6$ and $1.7 \times 10^5$. Express your answer in scientific notation.
Answer (1)
Convert both numbers to the same power of 10: 1.7 × 1 0 5 = 0.17 × 1 0 6 .
Add the numbers: ( 3.2 × 1 0 6 ) + ( 0.17 × 1 0 6 ) = ( 3.2 + 0.17 ) × 1 0 6 .
Calculate the sum: 3.2 + 0.17 = 3.37 .
Express the result in scientific notation: 3.37 × 1 0 6 .
Explanation
Understanding the Problem We are asked to find the sum of two numbers expressed in scientific notation and express the result in scientific notation as well. The given numbers are 3.2 × 1 0 6 and 1.7 × 1 0 5 .
Converting to the Same Power of 10 To add these two numbers, we need to express them with the same power of 10. We can rewrite 1.7 × 1 0 5 as 0.17 × 1 0 6 . This is because 1.7 × 1 0 5 = 1.7 × 1 0 5 × 10 10 = 0.17 × 1 0 6 .
Adding the Numbers Now we can add the two numbers: ( 3.2 × 1 0 6 ) + ( 0.17 × 1 0 6 ) = ( 3.2 + 0.17 ) × 1 0 6
Calculating the Sum We calculate the sum 3.2 + 0.17 = 3.37 .
Expressing the Result in Scientific Notation Therefore, the sum is 3.37 × 1 0 6 . Since 3.37 is between 1 and 10, this is already in scientific notation.
Final Answer Thus, the sum of 3.2 × 1 0 6 and 1.7 × 1 0 5 in scientific notation is 3.37 × 1 0 6 .
Examples
Scientific notation is extremely useful in many fields, especially in physics and astronomy. For example, the distance to the nearest star, Proxima Centauri, is approximately 4.017 × 1 0 16 meters. The mass of an electron is approximately 9.109 × 1 0 − 31 kilograms. Writing these numbers in standard decimal notation would be cumbersome and error-prone. Scientific notation provides a compact and convenient way to represent very large and very small numbers.
