Question

How do you simplify and write $\left( 1.25\times {{10}^{6}} \right)+\left( 250\times {{10}^{3}} \right)$ in standard notation?

Answer 1

Hint: To simplify $\left( 1.25\times {{10}^{6}} \right)+\left( 250\times {{10}^{3}} \right)$ , we have make the powers of 10 same. Let us consider $\left( 1.25\times {{10}^{6}} \right)$ . We can make the power of 10 as 3 by moving three places to the right. Let us split the power of 10. $\left( 1.25\times {{10}^{6}} \right)=\left( 1.25\times {{10}^{3}} \right)\times {{10}^{3}}=1250\times {{10}^{3}}$ .By substituting this in the given equation and simplifying, we will get the required answer. To convert the answer into standard form, we will append zeros which are equal to the power of 10, to the number part.

Complete step by step solution:
We need to simplify and write $\left( 1.25\times {{10}^{6}} \right)+\left( 250\times {{10}^{3}} \right)$ in standard notation. To simplify this, we have make the powers of 10 same. Let us consider $\left( 1.25\times {{10}^{6}} \right)$ . We can make the power of 10 as 3 by moving three places to the right. We can show this as follows.
Let us split the power of 10.
$\begin{align}
  & \left( 1.25\times {{10}^{6}} \right)=1.25\times {{10}^{3}}\times {{10}^{3}}=\left( 1.25\times {{10}^{3}} \right)\times {{10}^{3}} \\
 & =\left( 1.25\times 1000 \right)\times {{10}^{3}} \\
 & =1250\times {{10}^{3}} \\
\end{align}$
Now, we can easily simplify the given expression.
$\begin{align}
  & \left( 1.25\times {{10}^{6}} \right)+\left( 250\times {{10}^{3}} \right) \\
 & =\left( 1250\times {{10}^{3}} \right)+\left( 250\times {{10}^{3}} \right) \\
\end{align}$
Let us take ${{10}^{3}}$ outside since it is common to both the terms.
$\Rightarrow \left( 1250+250 \right)\times {{10}^{3}}$
On adding, we will get
$\Rightarrow 1500\times {{10}^{3}}$
Let us write this in standard notation, that is, in numbers. For this we will append zeros which are equal to the power of 10, to the number part. In other words, we will move decimal places to the right by the power of 10. Here, we will append 3 zeroes since the power of 10 is 3.
Hence, the required answer is 15,00,000 .

Note: When we move decimal places to right, we are multiplying it by 10 to the power of the number of decimal places in negative after moving to right. For example, $2.546=2546\times {{10}^{-3}}$ . To move decimal places to the left, we will multiply the number by 10 to the power of the number of decimal places after moving to the left. For example, to make 2569 to 2.569, we will multiply 2.569 by ${{10}^{3}}$ . That is, $2569=2.569\times {{10}^{3}}$ . When we want to make $1525\times {{10}^{4}}$ in scientific notation, we have to move 3 decimal places to the left so that we will get $1.525\times {{10}^{4}}\times {{10}^{3}}=1.525\times {{10}^{7}}$ .