Determine the quotient. Write your answer in scientific notation.
$\left(8.4 \times 10^6\right) \div\left(1.2 \times 10^2\right)$
Answer (1)
Divide the coefficients: 8.4/1.2 = 7 .
Divide the powers of 10: 1 0 6 /1 0 2 = 1 0 6 − 2 = 1 0 4 .
Combine the results: 7 t im es 1 0 4 .
The quotient in scientific notation is 7 t im es 1 0 4 .
Explanation
Understanding the problem We are asked to divide two numbers expressed in scientific notation and express the result in scientific notation as well. The given expression is ( 8.4 im es 1 0 6 ) d i v ( 1.2 t im es 1 0 2 ) .
Dividing the numbers To divide numbers in scientific notation, we divide the coefficients and divide the powers of 10. That is, f r a c 8.4 t im es 1 0 6 1.2 t im es 1 0 2 = f r a c 8.4 1.2 t im es f r a c 1 0 6 1 0 2
Dividing the coefficients First, let's divide the coefficients: f r a c 8.4 1.2 = 7
Dividing the powers of 10 Next, let's divide the powers of 10. Recall that when dividing exponential terms with the same base, we subtract the exponents: f r a c 1 0 6 1 0 2 = 1 0 6 − 2 = 1 0 4
Combining the results Now, we combine the results: f r a c 8.4 t im es 1 0 6 1.2 t im es 1 0 2 = 7 t im es 1 0 4
Final Answer The result is already in scientific notation, where the coefficient is 7 and the power of 10 is 1 0 4 . Therefore, the quotient is 7 t im es 1 0 4 .
Examples
Scientific notation is extremely useful in various fields like physics, astronomy, and engineering to represent very large or very small numbers. For example, the distance to the nearest star, Proxima Centauri, is approximately 4.017 t im es 1 0 13 kilometers. If we want to calculate how long it would take a spacecraft traveling at 1.0 t im es 1 0 5 km/h to reach Proxima Centauri, we would divide the distance by the speed: 1.0 t im es 1 0 5 4.017 t im es 1 0 13 = 4.017 t im es 1 0 8 hours. This shows how division with scientific notation simplifies calculations involving very large or small quantities, making it easier to understand and work with such numbers.
