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Comparision of Quantities Using Exponents

Solve for x: ...

Question

Solve for x:

$(a^{3 x + 5})^{2} . (a^{x})^{4} = a^{8 x + 12}$

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Solution

We have, $(a^{3 x + 5})^{2} . (a^{x})^{4} = a^{8 x + 12}$
$∴ (a^{6 x + 10}) . (a^{4 x}) = a^{8 x + 12}$
As base is same we can the powers,
$∴ a^{6 x + 10 + 4 x} = a^{8 x + 12}$
Now equating the powers, we get
$6 x + 10 + 4 x = 8 x + 12$
$2 x = 2$
$∴ x = 1$

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