Question

How do you divide and express your answer in scientific notation $\dfrac{8.8\times {{10}^{-8}}}{4\times {{10}^{-7}}}$?

Answer 1

Hint: We are given a fraction, $\dfrac{8.8\times {{10}^{-8}}}{4\times {{10}^{-7}}}$ and are asked to divide it and express the answer in scientific notation. To do so, we will learn how to divide the numbers when we have terms in scientific notation. We will learn about scientific notations and we will use the property, $\dfrac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}$, to get our answer. We will expand the exponential terms separately and the numbers separately and then solve them.

Complete step by step answer:
We are given a fraction (rational), $\dfrac{8.8\times {{10}^{-8}}}{4\times {{10}^{-7}}}$. Which means we have to divide $8.8\times {{10}^{-8}}$ by $4\times {{10}^{-7}}$. We can see that the numbers are given in scientific notations, so we will expand the terms, spare the exponential terms and separate the decimal numbers and then solve them separately.
Now, we have, $\dfrac{8.8\times {{10}^{-8}}}{4\times {{10}^{-7}}}$. We will separate it into decimal and exponential parts, so we get,
$\Rightarrow \dfrac{8.8}{4}\times \dfrac{{{10}^{-8}}}{{{10}^{-7}}}$
Now, we will solve them one by one. So, we have $\dfrac{8.8}{4}$, when we divide them, we get,
$4\overset{2.2}{\overline{\left){\begin{align}
  & 8.8 \\
 & 8 \\
 & \overline{\begin{align}
  & 08 \\
 & \text{ 8} \\
 & \overline{\text{ 0 }} \\
\end{align}} \\
\end{align}}\right.}}$
Hence, we get $\dfrac{8.8}{4}=2.2$.
Now, we will solve the exponential part, that is, $\dfrac{{{10}^{-8}}}{{{10}^{-7}}}$. We know the property that $\dfrac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}$. Using this here, we get, \[\dfrac{{{10}^{-8}}}{{{10}^{-7}}}={{10}^{\left( -8 \right)-\left( -7 \right)}}\]. Now, we will simplify the terms in the power. So, we know that –(-7) = 7, so we get, -8+7 which is equal to -1.
Hence, we get $\dfrac{{{10}^{-8}}}{{{10}^{-7}}}={{10}^{-1}}$.
We got that $\dfrac{8.8\times {{10}^{-8}}}{4\times {{10}^{-7}}}=\dfrac{8.8}{4}\times \dfrac{{{10}^{-8}}}{{{10}^{-7}}}=2.2\times {{10}^{-1}}$.
Now, we can see that in our answer, we have decimal after the first non-zero term and hence, it means that our answer is already in the scientific notation.
Therefore, the solution of $\dfrac{8.8\times {{10}^{-8}}}{4\times {{10}^{-7}}}=2.2\times {{10}^{-1}}$.

Note:
If we get the solution not in the scientific notation directly, then we have to convert it into the scientific notation. To do so, we will write the non-zero digit placing a decimal after the first non-zero digit. Then we will count the number of digits we need to move the original decimal to get it after a non-zero term, so we move decimal towards left, then we put positive power over the number 10 and if we move towards right, then a negative power is added on 10.