Given: \(\displaystyle{6}^{{{\frac{{{x}-{3}}}{{{4}}}}}}=\sqrt{{{6}}}\)

for solving this equation we make base same both side then equate power

we know that

\(\displaystyle{6}={\left(\sqrt{{{6}}}\right)}^{{2}}\)

so,

in place of 6 put \(\displaystyle{\left(\sqrt{{{6}}}\right)}^{{2}}\)

\(\displaystyle{\left({\left(\sqrt{{{6}}}\right)}^{{2}}\right)}^{{{\frac{{{x}-{3}}}{{{4}}}}}}=\sqrt{{{6}}}\)

\(\displaystyle{\left(\sqrt{{{6}}}\right)}^{{{\left({2}\times{\frac{{{x}-{3}}}{{{4}}}}\right)}}}=\sqrt{{{6}}}\)

\(\displaystyle{\left(\sqrt{{{6}}}\right)}^{{{\frac{{{x}-{3}}}{{{2}}}}}}={\left(\sqrt{{{6}}}\right)}^{{1}}\)

now here, base is same so power also be same

\(\displaystyle{\frac{{{x}-{3}}}{{{2}}}}={1}\)

\(\displaystyle{x}-{3}={2}\)

\(\displaystyle{x}={3}+{2}\)

\(\displaystyle{x}={5}\)

hence, solution of given expression is 5.

for solving this equation we make base same both side then equate power

we know that

\(\displaystyle{6}={\left(\sqrt{{{6}}}\right)}^{{2}}\)

so,

in place of 6 put \(\displaystyle{\left(\sqrt{{{6}}}\right)}^{{2}}\)

\(\displaystyle{\left({\left(\sqrt{{{6}}}\right)}^{{2}}\right)}^{{{\frac{{{x}-{3}}}{{{4}}}}}}=\sqrt{{{6}}}\)

\(\displaystyle{\left(\sqrt{{{6}}}\right)}^{{{\left({2}\times{\frac{{{x}-{3}}}{{{4}}}}\right)}}}=\sqrt{{{6}}}\)

\(\displaystyle{\left(\sqrt{{{6}}}\right)}^{{{\frac{{{x}-{3}}}{{{2}}}}}}={\left(\sqrt{{{6}}}\right)}^{{1}}\)

now here, base is same so power also be same

\(\displaystyle{\frac{{{x}-{3}}}{{{2}}}}={1}\)

\(\displaystyle{x}-{3}={2}\)

\(\displaystyle{x}={3}+{2}\)

\(\displaystyle{x}={5}\)

hence, solution of given expression is 5.