if $4^{4} + 4^{4} + 4^{4} + 4^{4} = 4^{a}$ , then $a =$ ___

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Solution

Given equation:
$4^{4} + 4^{4} + 4^{4} + 4^{4} = 4^{a}$
$\Rightarrow 4 \times 4^{4} = 4^{a}$
$\Rightarrow 4^{1} \times 4^{4} = 4^{a}$
$\Rightarrow 4^{4 + 1} = 4^{a}$ (since, Product rule: $a^{m} \times a^{n} = a^{m + n}$ )
$\Rightarrow 4^{5} = 4^{a}$
when the bases are equal then their powers are also
equal.
Therefore, $a = 5$ .

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