Express in scientific notation:
$\frac{\left(4 \times 10^5\right)\left(6 \times 10^{23}\right)}{1.2 \times 10^2}=$
A. $10 \times 10^{26}$
B. $20 \times 10^{26}$
C. $2 \times 10^{27}$
Answer (1)
Multiply the numbers and powers of 10 in the numerator: ( 4 × 1 0 5 ) ( 6 × 1 0 23 ) = 24 × 1 0 28 .
Divide the result by the denominator: 1.2 × 1 0 2 24 × 1 0 28 = 20 × 1 0 26 .
Convert to scientific notation: 20 × 1 0 26 = 2 × 1 0 1 × 1 0 26 = 2 × 1 0 27 .
The final answer is: 2 × 1 0 27 .
Explanation
Understanding the Problem We are asked to express the given expression in scientific notation and choose the correct answer from the given options. The expression is: 1.2 × 1 0 2 ( 4 × 1 0 5 ) ( 6 × 1 0 23 ) The available options are: 10 × 1 0 26 20 × 1 0 26 2 × 1 0 27
Simplifying the Numerator First, we multiply the numbers in the numerator: 4 × 6 = 24 Next, we multiply the powers of 10 in the numerator: 1 0 5 × 1 0 23 = 1 0 5 + 23 = 1 0 28 So, the numerator becomes: ( 4 × 1 0 5 ) ( 6 × 1 0 23 ) = 24 × 1 0 28
Dividing Numerator by Denominator Now, we divide the numerical part of the numerator by the numerical part of the denominator: 1.2 24 = 20 Next, we divide the powers of 10: 1 0 2 1 0 28 = 1 0 28 − 2 = 1 0 26 Combining the results, we get: 1.2 × 1 0 2 24 × 1 0 28 = 20 × 1 0 26
Expressing in Scientific Notation Finally, we express the result in scientific notation. Scientific notation requires the numerical part to be between 1 and 10. So we rewrite 20 as 2 × 1 0 1 :
20 × 1 0 26 = 2 × 1 0 1 × 1 0 26 = 2 × 1 0 1 + 26 = 2 × 1 0 27 Thus, the expression in scientific notation is 2 × 1 0 27 .
Choosing the Correct Answer Comparing our result with the given options, we find that the correct answer is 2 × 1 0 27 .
Examples
Scientific notation is extremely useful in various fields like physics, astronomy, and engineering where dealing with very large or very small numbers is common. For example, the distance to the Andromeda galaxy is approximately 2.5 × 1 0 22 meters. Similarly, the size of an atom is around 1 × 1 0 − 10 meters. Using scientific notation simplifies these numbers, making them easier to comprehend and use in calculations. This notation is also used in computer science to represent floating-point numbers.
