how are the graphs of the functions $f(x)=sqrt{16}^{x}$ and $g(x)=sqrt3{64}^{x}$ related?\nthe functions…

how are the graphs of the functions $f(x)=sqrt{16}^{x}$ and $g(x)=sqrt3{64}^{x}$ related?\nthe functions $f(x)$ and $g(x)$ are equivalent.\nthe function $g(x)$ increases at a faster rate.\nthe function $g(x)$ has a greater initial value.\nthe function $g(x)$ decreases at a faster rate.

how are the graphs of the functions $f(x)=sqrt{16}^{x}$ and $g(x)=sqrt3{64}^{x}$ related?\nthe functions $f(x)$ and $g(x)$ are equivalent.\nthe function $g(x)$ increases at a faster rate.\nthe function $g(x)$ has a greater initial value.\nthe function $g(x)$ decreases at a faster rate.

Answer

Explanation:

Step1: Simplify (f(x))

Rewrite (\sqrt{16^{x}}) as ((16^{x})^{\frac{1}{2}}). Using the power - of - a - power rule ((a^{m})^{n}=a^{mn}), we have ((16^{x})^{\frac{1}{2}} = 16^{\frac{x}{2}}). Since (16 = 2^{4}), then (16^{\frac{x}{2}}=(2^{4})^{\frac{x}{2}}=2^{2x}).

Step2: Simplify (g(x))

Rewrite (\sqrt[3]{64^{x}}) as ((64^{x})^{\frac{1}{3}}). Using the power - of - a - power rule ((a^{m})^{n}=a^{mn}), we get ((64^{x})^{\frac{1}{3}}=64^{\frac{x}{3}}). Since (64 = 2^{6}), then (64^{\frac{x}{3}}=(2^{6})^{\frac{x}{3}}=2^{2x}).

Step3: Compare the functions

Since (f(x)=2^{2x}) and (g(x)=2^{2x}), the two functions are equivalent.

Answer:

The functions (f(x)) and (g(x)) are equivalent.