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Laws of Exponents for Real Numbers

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Question

Solve : $2^{3 x + 1} + 2^{7} = (- 4)^{4}$

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Solution

$2^{3 x + 1} + 2^{7} = (- 4)^{4}$

$⟹ 2^{3 x + 1} + 2^{7} = (- 1)^{4} . (2^{2})^{4}$

$⟹ 2^{3 x + 1} + 2^{7} = 2^{8}$ (since, $(a^{m})^{n} = a^{m n}$ )

$⟹ 2^{3 x + 1} = 2^{8} - 2^{7}$

$⟹ 2^{3 x + 1} = 2^{7} (2 - 1)$

$⟹ 2^{3 x + 1} = 2^{7}$

when bases are equal then equate the powers

$⟹ 3 x + 1 = 7$

$⟹ 3 x = 6$

$⟹ x = 2$

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