Question

If (${\left( {\sqrt 6 } \right)^n} = 216$ then find the value of ${\left( {\text{n}} \right)^{\dfrac{3}{2}}}$

Answer 1

Hint: Proceed the solution by trying to make the same base on both sides so that we can compare the exponent of both sides to find the unknown variable.

Complete step-by-step answer:

\Here in the question it is given
⟹${\left( {\sqrt 6 } \right)^n} = 216$
We know that, square root can be write, raised to the power $\dfrac{1}{2}$
⟹${6^{\dfrac{{\text{n}}}{2}}} = {\text{ 216}}$
216 is the perfect cube of 6, so convert in cube form to simplify
⟹${6^{\dfrac{{\text{n}}}{2}}} = {{\text{6}}^3}$
Here, base 6 is same, so we can compare exponents
⟹$\dfrac{{\text{n}}}{2} = 3$
On cross-multiplication
⟹${\text{n}} = 6$
Hence we find the value of n.
In the question, it is asked the value of ${\left( {\text{n}} \right)^{\dfrac{3}{2}}}$
⟹${\left( {\text{n}} \right)^{\dfrac{3}{2}}} = {\left( 6 \right)^{\dfrac{3}{2}}}$ ; on putting n=6
⟹${\left( 6 \right)^{\dfrac{3}{2}}} = \sqrt {{6^3}} = 6\sqrt 6 $

Note: This type of particular problem, we can also solve by taking log on both sides, and on further solving using property on log, we can get to reach our answer.
⟹${\left( {\sqrt 6 } \right)^n} = 216$
Take log on both side
⟹$\log {\left( {\sqrt 6 } \right)^n} = \log 216$
 using property ${\log _{\text{b}}}({{\text{M}}^{\text{n}}}) = {\text{n}}{\log _{\text{b}}}({\text{M}})$
⟹$\dfrac{{\text{n}}}{2}\log (6) = \log (216)$
⟹$\dfrac{{\text{n}}}{2}\log (6) = \log {(6)^3}$
⟹$\dfrac{{\text{n}}}{2}\log (6) = 3\log (6)$
⟹$\dfrac{{\text{n}}}{2} = 3$
⟹${\text{n}} = 6$
⟹${\left( {\text{n}} \right)^{\dfrac{3}{2}}} = {\left( 6 \right)^{\dfrac{3}{2}}}$
⟹${\left( 6 \right)^{\dfrac{3}{2}}} = \sqrt {{6^3}} = 6\sqrt 6 $