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We are given two logarithmic equations: \[{\log _8}m = 3.5\] and ${\log _2}n = 7$.

We have a logarithmic identity: ${\log _a}x = b \Rightarrow x = {a^b}$. Using this identity in the given equations, we can calculate the values of m and n.

Using the identity in the equation: \[{\log _8}m = 3.5\], we get

\[ \Rightarrow {\log _8}m = 3.5 \Rightarrow m = {8^{3.5}}\]

Now, using the logarithmic identity in the equation: ${\log _2}n = 7$, we get

$ \Rightarrow {\log _2}n = 7 \Rightarrow n = {2^7}$

Now, when we have the values of m and n as: m = ${8^{3.5}}$and n =${2^7}$, we can write m = ${8^{3.5}}$ as $m = {8^{\dfrac{{35}}{{10}}}}$.

Now, we have $m = {8^{\dfrac{{35}}{{10}}}}$. It can be re – written as: $m = {\left( {{2^3}} \right)^{\dfrac{7}{2}}}$

Now, we can exchange the powers of 2 by using the algebraic property ${({x^a})^b} = {({x^b})^a}$.

$ \Rightarrow m = {\left( {{2^7}} \right)^{\dfrac{3}{2}}}$

Now, we know that the value of n = ${2^7}$, therefore, putting the value of n in the equation of m, we get

$ \Rightarrow m = {\left( n \right)^{\dfrac{3}{2}}}$

Or, we can write this as $m = n\sqrt n $.