Question

Which of the following is smallest?
E.$\sqrt[4] {5}$
F.$\sqrt[5] {4}$
G.$\sqrt 4 $
H.$\sqrt 3 $

Answer 1

Hint: Convert all the given numbers in the same format. Convert the roots to the same power, since we can only compare the numbers in the same system as we are given square-root, fourth root and fifth root.

Complete step-by-step answer:
Convert the given options in the same format.
Take the first number –
$\sqrt[4] {5}$
It can be written as –
$\sqrt[4] {5} = {5^{\dfrac{1}{4}}}$ .... (A)
Similarly equating the other given options –
$\sqrt[5] {4} = {4^{\dfrac{1}{5}}}$ .... (B)
$\sqrt 4 = {4^{\dfrac{1}{2}}}$ .... (C)
 $\sqrt 3 = {3^{\dfrac{1}{2}}}$ ..... (D)
Now take LCM (least common multiple) of the powers of all the above numbers.
We can observe that the least common multiple of the powers of all the terms is $\dfrac{1}{{20}}$ and therefore make all the powers equal accordingly.
Take equation (A)
$\sqrt[4] {5} = {5^{\dfrac{1}{4}}}$ -multiply and divide with $5$ in the power.
 \[ \Rightarrow {5^{\dfrac{1}{4} \times \dfrac{5}{5}}} = {5^{\dfrac{5}{1} \times \dfrac{1}{4} \times \dfrac{1}{5}}} = {\left( {{5^5}} \right)^{\dfrac{1}{{20}}}} = {\left( {3125} \right)^{\dfrac{1}{{20}}}}\] ..... (E)
Similarly,
Take equation (B)
$\sqrt[5] {4} = {4^{\dfrac{1}{5}}}$ -multiply and divide with $4$ in the power.
 \[ \Rightarrow {4^{\dfrac{1}{5} \times \dfrac{4}{4}}} = {4^{\dfrac{4}{1} \times \dfrac{1}{5} \times \dfrac{1}{4}}} = {\left( {{4^4}} \right)^{\dfrac{1}{{20}}}} = {\left( {256} \right)^{\dfrac{1}{{20}}}}\] .... (F)
Again, Similarly
Take equation (C)
$\sqrt 4 = {4^{\dfrac{1}{2}}}$ - multiply and divide with $10$ in the power.
 \[ \Rightarrow {4^{\dfrac{1}{2} \times \dfrac{{10}}{{10}}}} = {4^{\dfrac{{10}}{1} \times \dfrac{1}{2} \times \dfrac{1}{{10}}}} = {\left( {{4^{10}}} \right)^{\dfrac{1}{{20}}}} = {\left( {1048576} \right)^{\dfrac{1}{{20}}}}\] .... (G)
  Now, take Equation (D)
$\sqrt 3 = {3^{\dfrac{1}{2}}}$ - multiply and divide with $10$ in the power.
 \[ \Rightarrow {3^{\dfrac{1}{2} \times \dfrac{{10}}{{10}}}} = {3^{\dfrac{{10}}{1} \times \dfrac{1}{2} \times \dfrac{1}{{10}}}} = {\left( {{3^{10}}} \right)^{\dfrac{1}{{20}}}} = {\left( {59049} \right)^{\dfrac{1}{{20}}}}\] ...(H)
Now, comparing all the equations (E), (F), (G) and (H), we can observe that Equation (G) has the highest value.
So, the correct answer is “Option G”.

Note: Be good in multiples and be clear with the concepts of square roots, fourth roots and fifth roots. Also, refer to the laws of the power and exponents and know its identities. Always remember we can only compare numbers if they are in the same format.