Answer:
[tex]f(x)=4^{2x}[/tex]
Step-by-step explanation:
So we are given:
[tex]f(x)=(8^{\frac{2}{3}x})(16^{\frac{1}{2}x})[/tex]
and we have to find a function that will produce a similar graph. To do this, lets first simplify the given function.
Here are some properties of exponents that I will be using:
[tex]c^{\frac{n}{x}}=\sqrt[x]{c^{n}}=(\sqrt[x]{c})^{n}[/tex]
[tex](c^{x})^{n}=(c^{n})^{x}=c^{xn}[/tex]
[tex]c^{x}*c^{n}=c^{x+n}[/tex]
Now lets begin simplifying the function.
[tex]f(x)=(8^{\frac{2}{3}x})(16^{\frac{1}{2}x})[/tex]
[tex]=((\sqrt[3]{8^{2}})^{x})((\sqrt{16})^{x})[/tex]
[tex]=(4^{x})(4^{x})[/tex]
[tex]=4^{2x}[/tex]
So the correct answer would be [tex]f(x)=4^{2x}[/tex].
I hope you find my answer and explanation to be helpful. Happy studying.
