What is the numerical value of $\frac{\left(3.00 \times 10^6\right)\left(2.0 \times 10^{-3}\right)}{5.0 \times 10^{-2}} ?$
A. $1.2 \times 10^7$
B. $1.2 \times 10^3$
C. $1.2 \times 10^{11}$
D. $1.2 \times 10^5$
E. 1.2
Answer (1)
Multiply the numbers in the numerator: 3.00 × 2.0 = 6.00 .
Multiply the powers of 10 in the numerator: 1 0 6 × 1 0 − 3 = 1 0 6 + ( − 3 ) = 1 0 3 .
Divide the numerical part of the numerator by the numerical part of the denominator: 5.0 6.00 = 1.2 .
Divide the powers of 10: 1 0 − 2 1 0 3 = 1 0 3 − ( − 2 ) = 1 0 3 + 2 = 1 0 5 . The final result is 1.2 × 1 0 5 .
Explanation
Understanding the Problem We are asked to find the numerical value of the expression 5.0 × 1 0 − 2 ( 3.00 × 1 0 6 ) ( 2.0 × 1 0 − 3 ) . This involves multiplying and dividing numbers expressed in scientific notation.
Multiplying Numerator Values First, let's multiply the numbers in the numerator: 3.00 × 2.0 = 6.00 .
Multiplying Numerator Powers of 10 Next, multiply the powers of 10 in the numerator: 1 0 6 × 1 0 − 3 = 1 0 6 + ( − 3 ) = 1 0 3 .
Simplified Numerator So, the numerator simplifies to 6.00 × 1 0 3 .
Dividing Numerical Values Now, divide the numerical part of the numerator by the numerical part of the denominator: 5.0 6.00 = 1.2 .
Dividing Powers of 10 Divide the powers of 10: 1 0 − 2 1 0 3 = 1 0 3 − ( − 2 ) = 1 0 3 + 2 = 1 0 5 .
Final Result Therefore, the result is 1.2 × 1 0 5 .
Examples
Scientific notation is used in many fields, such as physics and astronomy, to represent very large or very small numbers. For example, the distance to a star might be 4.5 × 1 0 16 meters. Calculations involving such numbers are simplified using the rules of exponents. Understanding scientific notation helps in expressing and manipulating these quantities efficiently.
