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[tex] \sqrt{4 \sqrt{4 \sqrt{4} } } = \sqrt{4 \sqrt{ {4}^{1 + \frac{ 1}{2} } } } = \sqrt{4 \sqrt{ {4}^{ \frac{3}{2} } } } = \sqrt{4 \times {4}^{ \frac{3}{4} } } = \sqrt{ {4}^{ \frac{7}{4} } } = {4}^{ \frac{7}{8} } = {2}^{ \frac{7}{4} } [/tex]