Question

How do you write $0.5\times {{10}^{-4}}$ in expanded form?

Answer 1

Hint: This type of question needs the concept of expansion. To solve this question we need to convert the number with power format into the product of the base (base is the number which has power on it ). Here ${{10}^{-4}}$ will be changed to the product format ${{a}^{-n}}=\dfrac{1}{{{a}^{n}}}$ , where a is 10 and n is 4. Then we can multiply it with 0.5 to get the expanded form.

Complete step by step solution:
We are supposed to expand $0.5\times {{10}^{-4}}$ , therefore we have to remove the term ${{10}^{-4}}$ from the expression.
${{10}^{^{-4}}}$ can be removed from the expression by the following process
We know that ${{a}^{-n}}$ can be represented as ${{a}^{-n}}=\dfrac{1}{{{a}^{n}}}$ which is further represented by $\dfrac{1}{a\times a\times a........(ntimes)}$ , for the cases when n is a natural number.
So considering $10$ as “a” and $4$ as “n”, we get :
${{10}^{-4}}=\dfrac{1}{{{10}^{4}}}$
We will change the above number in product format
\[{{10}^{-4}}=\dfrac{1}{10000}\]
Now changing the fraction in decimal form format , we get
$\Rightarrow 0.0001$
Now for calculating , we will multiply the expanded fraction (now decimal number) to $0.5$ :
 \[\begin{align}
  & \Rightarrow \text{0}\text{.5}\times 0.0001 \\
 & \Rightarrow 0.00005 \\
\end{align}\]
$\therefore $ The expanded form of $0.5\times {{10}^{-4}}$ is \[0.00005\] .

Note: The above solving is a bit longer so instead we can have another approach to solve this question. Instead of solving the equation and writing in the fraction form , we can directly see the power of 10 and change the decimal place of the number accordingly.
Here , \[0.5\times {{10}^{-4}}=0.00005\]
Since the power of 10 is -4 , the place of the decimal moves to 4 digits left to the current position of the decimal . Therefore the current position of decimal in $0.5$ is left to one digit , now since the power is $-4$ the decimal point moves towards left crossing $4$ digits . Therefore the value results in \[0.00005\].