1. **State the problem:** We are given functions \(g(x) = x^4\), \(h(x) = 3^x\), and \(f(x) = 7 + 3x\). We need to find the value of \(k\) such that \(h(3x) = k^x\). 2. **Write the given equation:** $$h(3x) = k^x$$ Since \(h(x) = 3^x\), substitute \(3x\) into \(h\): $$h(3x) = 3^{3x}$$ 3. **Set the expressions equal:** $$3^{3x} = k^x$$ 4. **Rewrite the right side:** $$k^x = (k)^x$$ 5. **Since the bases are raised to the power \(x\), equate the bases:** $$3^{3} = k$$ 6. **Calculate \(k\):** $$k = 3^3 = 27$$ **Final answer:** $$\boxed{27}$$