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