Question

How do you write $0.0045$ in scientific notation? \[\]

Answer 1

Hint: We recall the scientific notation or standard form of writing number where we ensure that exactly one non-zero digit lies at the left side of the number by multiplying power of 10 that is ${{10}^{n}}$. We move the first non-zero digit 4 of $0.0045$ to the left of decimal point and the number of places moved will be $n$. \[\]

Complete step-by-step solution:
We know that scientific notation of a number commonly represented in the form
\[m\times {{10}^{n}}\]
Here the exponent $n$ is always an integer and the number $m$ called as significand is a non-zero real number whose absolute value $\left| m \right|$ always lie between 1 and 10 $\left( 1\le \left| m \right| < 10 \right)$. So we have to ensure only one non-zero digit lies at the left of the decimal point which can be either in positive or negative sign. \[\]
We know that if we move decimal point of number toward right by $n$ places then we divide the number by ${{10}^{n}}$ and if we move decimal point towards left by $n$ places then we multiply by ${{10}^{n}}$ with the number. \[\]
We are asked to find the scientific notation of $0.0045$. We see that decimal number 4 is the first non-zero digit which is 3 places towards the right of the decimal point. So we move 4 towards left by 3 places and write the number as $4.5$. Since we moved the decimal point by 3 places towards right we have to divide by ${{10}^{3}}$ which means
\[0.0045=\dfrac{4.5}{{{10}^{3}}}=4.5\times \dfrac{1}{{{10}^{3}}}\]
We know from laws of exponent that $\dfrac{1}{{{a}^{p}}}={{a}^{-p}}\left( a\ne 0 \right)$. So we have the scientific notation as
\[0.0045=\dfrac{4.5}{{{10}^{3}}}=4.5\times \dfrac{1}{{{10}^{3}}}=4.5\times {{10}^{-3}}\]

Note: We know that standard form or scientific notation of a number is used when we cannot write a very small or very large number decimal form. The example of smaller numbers like measure of wavelength in Armstrong $\left( {{10}^{-8}} \right)$ or nanometre $\left( {{10}^{-9}} \right)$ . The example of large numbers is the weight of plants like the weight of earth is $5.972\times {{10}^{24}}$ kg.