This is a common question that students have but are ashamed to ask.

As per your logic, $1mm={10}^{-3}m$.

Multiplying both sides by 2 to maintain equality we get

$2\times 1mm=2\times {10}^{-3}m$ and not ${10}^{-6}mm$.

Please recall the property of exponents which is as follows....

${a}^{m}\times {a}^{n}={a}^{\left(m+n\right)}$ therefore, ${a}^{-m}\times {a}^{-n}={a}^{\left(-m-n\right)}$

Applying this property in our example,

${10}^{-6}={10}^{-3}\times {10}^{-3}$=${10}^{\left(-3-3\right)}={10}^{-6}$

So,

$\begin{array}{l}{10}^{-6}m={10}^{-6}\times 1m\\ ={10}^{-6}\times {10}^{+3}mm\left[::1m=1000mm\right]\end{array}$

${10}^{-6}m={10}^{\left(-6+3\right)}mm\phantom{\rule{0ex}{0ex}}$

$\begin{array}{l}{10}^{-6}m={10}^{-3}mm\\ =\frac{1}{{10}^{3}}mm=\frac{1}{1000}mm\end{array}$

Anand Kurien

AuthorH.S. Mani

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