Question

Which of the following is false?
A. $\sqrt[5]{6} < \sqrt[4]{5}$
B. $\sqrt[9]{4} < \sqrt[6]{3}$
C. $\sqrt[3]{{\dfrac{1}{3}}} < \sqrt[2]{{\dfrac{1}{2}}}$
D. $\sqrt[3]{2} < \sqrt[6]{3}$

Answer 1

Hint: To find the ${n^{th}}$ root of any real number, take both the terms in the expression to the power that is common to the root given in the expression. A root can be defined as the number that is multiplied by itself the root number of times. The ${n^{th}}$ root is used $n$ times in multiplication to get the original value.

Formula used: The following formulae can be used to solve these questions:
${({x^m})^n} = {x^{mn}}$
${(\sqrt[n]{x})^n} = x$

Complete step by step solution:
Considering the first option, $\sqrt[5]{6} < \sqrt[4]{5}$
For $\sqrt[5]{6}$ , raise the term to the power of $\;20$ , to get ${(\sqrt[5]{6})^{20}}$ ,
Now, applying the laws of exponents ${({x^m})^n} = {x^{mn}}$ to the above expression we get,
${(\sqrt[5]{6})^{20}} = {({(\sqrt[5]{6})^5})^4}$
Further on simplification, we get,
${({(\sqrt[5]{6})^5})^4} = {(6)^4}$
$\Rightarrow {(6)^4} = 1296$
Similarly, for $\sqrt[4]{5}$ after raising the term to the power of $\;20$ we get,
${(\sqrt[4]{5})^{20}} = {({(\sqrt[4]{5})^4})^5}$
Simplifying the above expression, we get,
${({(\sqrt[4]{5})^4})^5} = {(5)^5}$
${(5)^5} = 3125$
Hence, it is clear that $\sqrt[5]{6} < \sqrt[4]{5}$ . Since, $1296 < 3125$ therefore, this option is true.
Considering the second option, $\sqrt[9]{4} < \sqrt[6]{3}$
Raising both sides of the inequality to the power of $\;18$ , we get
${\left( {\sqrt[9]{4}} \right)^{18}} < {\left( {\sqrt[6]{3}} \right)^{18}}$
Taking Left Hand Side ${\left( {\sqrt[9]{4}} \right)^{9 \times 2}}$ ;
$= {4^2}$
$= 16$
Taking Right Hand Side ${\left( {\sqrt[6]{3}} \right)^{6 \times 3}}$ ;
$= {3^3}$
$= 27$
Comparing results we find $16 < 27$
$\Rightarrow \sqrt[9]{4} < \sqrt[6]{3}$
Therefore this option is true.
Considering the third option, $\sqrt[3]{{\dfrac{1}{3}}} < \sqrt[2]{{\dfrac{1}{2}}}$
Raising both side of inequality to the power of $6$ ;
Left Hand Side: ${\left( {\sqrt[3]{{\dfrac{1}{3}}}} \right)^6}$
$\Rightarrow {\left( {\sqrt[3]{{\dfrac{1}{3}}}} \right)^{3 \times 2}}$
$= {\left( {\dfrac{1}{3}} \right)^2}$
Simplifying the square,
$= \dfrac{1}{9}$
Taking Right Hand Side: ${\left( {\sqrt[2]{{\dfrac{1}{2}}}} \right)^6}$
$\Rightarrow {\left( {\sqrt[2]{{\dfrac{1}{2}}}} \right)^{3 \times 2}}$
$= {\left( {\dfrac{1}{2}} \right)^3}$
Simplifying the power of cube
$= \dfrac{1}{8}$
Comparing L.H.S with R.H.S, we get $\dfrac{1}{9} < \dfrac{1}{8}$
$\Rightarrow \sqrt[3]{{\dfrac{1}{3}}} < \sqrt[2]{{\dfrac{1}{2}}}$
Therefore this option is correct.
Considering the fourth option, $\sqrt[3]{2} < \sqrt[6]{3}$
Raising both side of the inequality to the power$6$;
Left Hand Side ${\left( {\sqrt[3]{2}} \right)^6}$ ;
$= {\left( {\sqrt[3]{2}} \right)^{2 \times 3}}$
$= {2^2}$
Simplifying the power,
$= 4$
Taking Right Hand Side ${\left( {\sqrt[6]{3}} \right)^6}$ ;
$= {3^1}$
$= 3$
Comparing L.H.S and R.H.S. $4 > 3$ ,
$\Rightarrow \sqrt[3]{2} > \sqrt[6]{3}$

Therefore, the false option from the given options is, D.

Note: There are many ways to find the ${n^{th}}$ root of a given real number. The most commonly used is the trial and error and the multiplication method used. Roots can also include decimal numbers and fractions.
Always take a number one greater and one lesser than the first number while trying the trial and error method to find the ${n^{th}}$ root. Then, take the number that is the closest to the approximate value of the root value.