Question

How do you simplify $1000000000\times 6000000000$?

Answer 1

Hint: We first convert the standard forms of the given terms to their scientific forms to simplify. Then we complete the multiplication using the concept of indices. The addition of the indices gives the final solution.

Complete step by step solution:
To simplify the multiplication, we first convert the standard forms of the given terms to their scientific forms.
We move the decimal to the smallest one-digit place so that it can be expressed in the multiplication form of 10. For $1000000000$ and $6000000000$, we keep moving the decimal to the left. The decimal starts after the last zero in both cases. We multiply with 10 to compensate for the movement.
In $1000000000$, the movement of the decimal point happens 9 times which means \[1000000000=1\times {{10}^{9}}\].
In $6000000000$, the movement of the decimal point happens 9 times which means \[6000000000=6\times {{10}^{9}}\].
Now we apply the concept of indices to complete the multiplication.
We have \[\left( 1\times {{10}^{9}} \right)\times \left( 6\times {{10}^{9}} \right)=6\times {{10}^{9}}\times {{10}^{9}}\].
We know the exponent form of the number $a$ with the exponent being $n$ can be expressed as ${{a}^{n}}$.
We take two exponential expressions where the exponents are $m$ and $n$.
Let the numbers be ${{a}^{m}}$ and ${{a}^{n}}$. We take multiplication of these numbers.
The indices get added. So, ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$.
Therefore, \[6\times {{10}^{9}}\times {{10}^{9}}=6\times {{10}^{9+9}}=6\times {{10}^{18}}\].
The simplified form of $1000000000\times 6000000000$ is \[6\times {{10}^{18}}\].

The simplified form of $1000000000\times 6000000000$ is \[6\times {{10}^{18}}\].

Note: We have to remember that the multiplication of the indices is possible only when the base is equal. We can also expand the scientific form to its standard form. But that will be tough to understand.