# Solve the exponential equation 6^{(x-3)/4}=\sqrt{6} by expressing each sid

Solve the exponential equation $$\displaystyle{6}^{{\frac{{{x}-{3}}}{{4}}}}=\sqrt{{{6}}}$$ by expressing each side as a power of the same base and then equating exponents.

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doplovif
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.