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