Answer: k = 6+10x
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Work Shown:
[tex]64*4^{5x}\\\\\left(2^6\right)*\left(2^{2}\right)^{5x}\\\\\left(2^6\right)*\left(2^{2*5x}\right)\\\\\left(2^6\right)*\left(2^{10x}\right)\\\\2^{6+10x}\\\\[/tex]
The result is in the form [tex]2^k[/tex] with k = 6+10x
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Explanation:
First we need to get everything as an exponential expression with a base 2.
We rewrite 64 as [tex]2^6[/tex] and 4 as [tex]2^2[/tex] in the second step.
In the third step, I used the rule [tex](a^b)^c = a^{b*c}[/tex] which says to multiply the exponents together. That's how I went from [tex](2^2)^{5x}[/tex] to [tex]2^{2*5x}[/tex]. Then the 2*5x in the exponent becomes 10x.
To wrap things up, I used the rule [tex]a^b*a^c = a^{b+c}[/tex] which says to add the exponents when we multiply stuff of the same base together.
