Given that \(a^3a^k = a^9\) for all real numbers \(a\), what is the value of \(k\) ?

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We use the product rule for exponents:

$$ x^a\cdot x^b = x^{a+b} $$

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$$ a^3 a^k = a^{3+k} = a^9 $$ $$ 3+k = 9 $$ $$ k = \boxed{6} $$