Question

How do you express $5\times {{10}^{-4}}$ in standard notation?

Answer 1

Hint: The notation of the number given in the above question, which is $5\times {{10}^{-4}}$, is the scientific notation since it is written as a multiple of the power of ten. For writing it in the standard notation, we have to use the property of the negative exponents, given by ${{a}^{-m}}=\dfrac{1}{{{a}^{m}}}$ so that the given number can be written as $\dfrac{5}{{{10}^{4}}}$. The fraction $\dfrac{5}{{{10}^{4}}}$ means that we have to divide $5$ four times by $10$. We know that the division of a number by ten shifts its decimal point one unit to the left. So the division of $5$ by ${{10}^{4}}$ will shift its decimal point four units to the left, and the decimal number obtained will be the required standard notation of the given number.

Complete step-by-step solution:
Let us write the number given in the above question as
$\Rightarrow n=5\times {{10}^{-4}}$
We can observe that the given number is expressed in the form of scientific notation. For expressing it in the standard form, we have to write it in the form of decimal. For this, we use the negative exponent property which is given by ${{a}^{-m}}=\dfrac{1}{{{a}^{m}}}$, to write \[{{10}^{-4}}\] as \[\dfrac{1}{{{10}^{4}}}\] in the above number to get
$\Rightarrow n=\dfrac{5}{{{10}^{4}}}$
The above expression tells us that we have to divide $5$ four times by $10$. We know that the division of a number by ten shifts its decimal point one unit to the left. So by dividing $5$ four times by $10$, its decimal point four units to the left so that the above fraction will become
$\Rightarrow n=0.0005$
Hence, the given number $5\times {{10}^{-4}}$ is written in the standard form as $0.0005$.

Note: For the case of the numbers expressed in the form of the scientific notation, in which the exponent over ten is negative, we must check that the number of zeroes before the first non-zero digit in their standard form must be equal to the absolute value of the exponent. For example, in the above solution, we can check that the number of zeroes in $0.0005$ is equal to four, which is equal to the absolute value of the exponent in $5\times {{10}^{-4}}$.