Question

If 1 mg \[n{s^{ - 1}} = {10^x}\mu gp{s^{ - 1}}\], then the value of x is.
(A) 1
(B) 2
(C) -1
(D) 0

Answer 1

Hint Write all the units on the left hand side and right-hand side in terms of the S.I unit system that is grams (g) and seconds (s) by converting the prefixes into g and s. Now, using the powers rules, powers are equal when bases are equal giving the value of x.

Complete step by step answer:
Convert the units in terms of grams and seconds
LHS:
 $
  1mg = {10^{ - 3}}g \\
  1ns = {10^{ - 9}}s \\
 $
RHS:
 $
  1\mu g = {10^{ - 6}}g \\
  1ps = {10^{ - 12}}s \\
 $
So, on equating we get
 $
  \dfrac{{{{10}^{ - 3}}g}}{{{{10}^{ - 9}}s}} = \dfrac{{{{10}^x} \times {{10}^{ - 6}}g}}{{{{10}^{ - 12}}s}} \\
  {10^{ - 3 + 9}}g{s^{ - 1}} = {10^{x - 6 + 12}}g{s^{ - 1}} \\
  {10^6}g{s^{ - 1}} = {10^{x + 6}}g{s^{ - 1}} \\
 $
Since bases are equal on both sides the powers should also be equal
On equating the powers, we get
 $
  6 = x + 6 \\
  x = 0 \\
 $

Hence the value of x is zero and the correct option above is D.

Note: Units of measurements can express very large quantity and very small quantities as well using some prefixes given below

	S I Prefixes
Multiple	Prefix	Symbol
\[\begin{array}{*{20}{c}} {{{10}^{15}}} \end{array}\]	peta	P
\[\begin{array}{*{20}{c}} {{{10}^{12}}} \end{array}\]	tera	T
\[\begin{array}{*{20}{c}} {{{10}^9}} \end{array}\]	giga	G
\[\begin{array}{*{20}{c}} {{{10}^6}} \end{array}\]	mega	M
\[\begin{array}{*{20}{c}} {{{10}^3}} \end{array}\]	kilo	k
\[\begin{array}{*{20}{c}} {{{10}^2}} \end{array}\]	hecto	h
10	deka	da
\[{10^{ - 1}}\]	deci	d
\[\begin{array}{*{20}{c}} {{{10}^{ - 2}}} \end{array}\]	centi	c
\[\begin{array}{*{20}{c}} {{{10}^{ - 3}}} \end{array}\]	milli	M
\[\begin{array}{*{20}{c}} {{{10}^{ - 6}}} \end{array}\]	micro	\[\begin{array}{*{20}{c}} \mu \end{array}\]
\[\begin{array}{*{20}{c}} {{{10}^{ - 9}}} \end{array}\]	nano	n
\[\begin{array}{*{20}{c}} {{{10}^{ - 12}}} \end{array}\]	pico	p
\[\begin{array}{*{20}{c}} {{{10}^{ - 15}}} \end{array}\]	femto	f