Question

In scientific notation, \[670,000,000 + 700,000,000 = \]?
A. \[1.37 \times {10^{ - 9}}\]
B. \[1.37 \times {10^7}\]
C. \[1.37 \times {10^8}\]
D. \[1.37 \times {10^9}\]
E. \[1.37 \times {10^{15}}\]

Answer 1

Hint: In this question we convert the question into a form where all the zero digits behind the non-zero number are collected as a power of ten, and then we add the non-zero number multiplied with the power of collection of tens.
*A number with \[n\] zero digits after the non-zero number say \[p,000,000\] will be written as \[p \times {10^6}\]
Where we multiplied with \[{10^6}\] because there are six zero digits after the non-zero number \[p\].

Complete step-by-step answer:
We are given scientific notation as \[670,000,000 + 700,000,000\]
We solve for the two numbers separately.
First we solve \[670,000,000\]
Here we count the number of zero digits is \[7\] and the non-zero number is \[67\]
Therefore substituting from the formula we can write
\[670,000,000 = 67 \times {10^7}\]
Now we solve \[700,000,000\]
Here we count the number of zero digits is \[8\] and the non-zero number is \[7\]
But since we have our first number with \[{10^7}\], therefore we try to make a second number also with the same power so that when we add two numbers we don’t get confused.
So, we take number of zero digits is \[7\]and the non-zero number is \[70\]
Therefore substituting from the formula we can write
\[700,000,000 = 70 \times {10^7}\]
Now we substitute both the values in our equation and add the numbers by taking common \[{10^7}\]
\[670,000,000 + 700,000,000 = 67 \times {10^7} + 70 \times {10^7}\]
Take \[{10^7}\]common from both the numbers on the RHS of the equation.
\[670,000,000 + 700,000,000 = (67 + 70) \times {10^7}\]
Adding the values inside the bracket.
\[670,000,000 + 700,000,000 = (137) \times {10^7}\]
Now looking at the options given in the question we see the non-zero number is \[1.37\] i.e. decimal up to two numbers.
We can make the term \[137\] into \[1.37\] by dividing and multiplying by hundred because it contains two zero digits and we can write any decimal number up to two digits by dividing it from hundred.
Therefore, multiply and divide\[137\] by \[100\]
\[
  137 = 137 \times \dfrac{{100}}{{100}} \\
  137 = \dfrac{{137}}{{100}} \times 100 \\
 \]
Now we write \[\dfrac{{137}}{{100}} = 1.37\] and collecting the zero digits after one, there are two digits, so \[100 = {10^2}\]
\[137 = 1.37 \times {10^2}\]
Now we substitute the value of \[137 = 1.37 \times {10^2}\] in the equation \[670,000,000 + 700,000,000 = (137) \times {10^7}\]
\[670,000,000 + 700,000,000 = 1.37 \times {10^2} \times {10^7}\]
Now we know from the property of exponents that when the base is same then powers can be added, i.e. \[{p^m} \times {p^n} = {p^{m + n}}\]
So, \[{10^7} \times {10^2} = {10^{7 + 2}} = {10^9}\]
Therefore, after the substitution of \[{10^7} \times {10^2} = {10^9}\]the number becomes
 \[670,000,000 + 700,000,000 = 1.37 \times {10^9}\]

So, the correct answer is “Option D”.

Note: Students are likely to make mistakes if they don’t convert both the numbers with the same multiple of ten with the same number in power which leads to error in calculation.