Determine the difference. Write your answer in scientific notation.
$3.4 \times 10^{-2}-7.6 \times 10^{-4}$
Answer (1)
Convert both numbers to the same power of 10: 3.4 × 1 0 − 2 = 340 × 1 0 − 4 .
Subtract the numbers: 340 × 1 0 − 4 − 7.6 × 1 0 − 4 = 332.4 × 1 0 − 4 .
Convert the result to scientific notation: 332.4 × 1 0 − 4 = 3.324 × 1 0 − 2 .
The difference in scientific notation is 3.324 × 1 0 − 2 .
Explanation
Understanding the Problem We are asked to find the difference between 3.4 × 1 0 − 2 and 7.6 × 1 0 − 4 . The answer should be written in scientific notation.
Converting to the Same Power of 10 To subtract these numbers, we need to express them with the same power of 10. We can rewrite 3.4 × 1 0 − 2 as 3.4 × 1 0 − 2 = 3.4 × 1 0 2 × 1 0 − 4 = 340 × 1 0 − 4 .
Subtracting the Numbers Now we can subtract: 340 × 1 0 − 4 − 7.6 × 1 0 − 4 = ( 340 − 7.6 ) × 1 0 − 4 .
Calculating the Difference Calculating the difference inside the parentheses: 340 − 7.6 = 332.4 . So we have 332.4 × 1 0 − 4 .
Converting to Scientific Notation Now we convert 332.4 × 1 0 − 4 to scientific notation. To do this, we need to write the number as a × 1 0 b , where 1 ≤ a < 10 and b is an integer. We can rewrite 332.4 × 1 0 − 4 as 3.324 × 1 0 2 × 1 0 − 4 = 3.324 × 1 0 2 − 4 = 3.324 × 1 0 − 2 .
Final Answer Therefore, the difference is 3.324 × 1 0 − 2 .
Examples
Scientific notation is extremely useful in fields like physics and astronomy, where dealing with very large or very small numbers is common. For example, the speed of light is approximately 3.0 × 1 0 8 meters per second, and the mass of an electron is approximately 9.11 × 1 0 − 31 kilograms. Using scientific notation makes these numbers easier to work with and understand. This notation also helps in calculating distances between stars or the size of atoms, making complex calculations more manageable.
