Add $\left(7.8 \times 10^5\right)+\left(2.4 \times 10^5\right)$.
A) $10.2 \times 10^6$
B) $10.2 \times 10^{10}$
C) $1.02 \times 10^6$
D) $1.02 \times 10^5$
Answer (2)
Factor out 1 0 5 : ( 7.8 × 1 0 5 ) + ( 2.4 × 1 0 5 ) = ( 7.8 + 2.4 ) × 1 0 5 .
Add the coefficients: 7.8 + 2.4 = 10.2 , so we have 10.2 × 1 0 5 .
Convert to scientific notation: 10.2 × 1 0 5 = ( 1.02 × 1 0 1 ) × 1 0 5 = 1.02 × 1 0 6 .
The final answer is: 1.02 × 1 0 6 .
Explanation
Problem Analysis We are asked to add two numbers expressed in scientific notation: ( 7.8 × 1 0 5 ) + ( 2.4 × 1 0 5 ) . Our goal is to find the sum and match it to one of the given options.
Factoring and Simplifying First, notice that both terms have the same power of 10, which is 1 0 5 . This allows us to factor out 1 0 5 and add the coefficients directly: ( 7.8 × 1 0 5 ) + ( 2.4 × 1 0 5 ) = ( 7.8 + 2.4 ) × 1 0 5
Adding the Coefficients Now, we add the coefficients 7.8 and 2.4: 7.8 + 2.4 = 10.2
Combining the Results So, the expression becomes: 10.2 × 1 0 5
Converting to Scientific Notation To express this in proper scientific notation, we need to write 10.2 as 1.02 × 1 0 1 . Therefore, 10.2 × 1 0 5 = ( 1.02 × 1 0 1 ) × 1 0 5
Final Calculation Using the property of exponents, we add the powers of 10: 1.02 × 1 0 1 × 1 0 5 = 1.02 × 1 0 1 + 5 = 1.02 × 1 0 6
Selecting the Correct Option Comparing our result, 1.02 × 1 0 6 , with the given options, we see that it matches option C. Therefore, the correct answer is C) 1.02 × 1 0 6 .
Examples
Scientific notation is used in many real-world applications, such as expressing very large numbers like the distance between stars or very small numbers like the size of an atom. For example, the distance to the nearest star, Proxima Centauri, is approximately 4.017 × 1 0 13 kilometers. Similarly, the size of a hydrogen atom is about 5.3 × 1 0 − 11 meters. Using scientific notation makes it easier to work with these numbers and perform calculations.
By adding the coefficients of the numbers in scientific notation, we found that ( 7.8 × 1 0 5 ) + ( 2.4 × 1 0 5 ) = 10.2 × 1 0 5 . Converting to proper scientific notation gives us 1.02 × 1 0 6 , which is option C. Thus, the answer is C) 1.02 × 1 0 6 .
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