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Exponents with like Bases

Solve for x...

Question

Solve for $x$ : $(a^{3 x + 5})^{2} \cdot (a^{x})^{4} = a^{8 x + 12}$

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Solution

Given, $(a^{3 x + 5})^{2} . (a^{x})^{4} = a^{8 x + 12}$
$\Rightarrow (a^{6 x + 10}) . (a^{4 x}) = a^{8 x + 12}$
$\Rightarrow a^{6 x + 10 + 4 x} = a^{8 x + 12}$
As the bases are equal, powers must be equal.
Therefore, $6 x + 10 + 4 x = 8 x + 12$
$\Rightarrow 2 x = 2$
$\Rightarrow x = 1$

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