\sqrt[3]{\sqrt[4]{\left(5-2\right)^2-2^8\cdot \:2^{-6}}}+\frac{1}{2}\cdot \:\sqrt{3}\cdot \:\sqrt{27}-\sqrt[3]{8}

asked by guest
on Nov 25, 2024 at 6:23 pm



You asked:

Evaluate the expression: \(\sqrt[3]{\sqrt[4]{{\left( 5 - 2 \right)}^{2} - {2}^{8} {2}^{-6}}} + \frac{1}{2} \cdot \sqrt{3} \cdot \sqrt{27} - \sqrt[3]{8}\)

MathBot Answer:

\[\sqrt[3]{\sqrt[4]{{\left( 5 - 2 \right)}^{2} - {2}^{8} {2}^{-6}}} + \frac{1}{2} \cdot \sqrt{3} \cdot \sqrt{27} - \sqrt[3]{8} = \sqrt[12]{5} + \frac{5}{2} \approx 3.64352983608292030922678781625508\]

\[\begin{aligned}\sqrt[3]{\sqrt[4]{{\left( 5 - 2 \right)}^{2} - {2}^{8} {2}^{-6}}} + \frac{1}{2} \cdot \sqrt{3} \cdot \sqrt{27} - \sqrt[3]{8}& = \sqrt[3]{\sqrt[4]{\left(5 - 2\right)^{2} - 2^{8} \cdot \frac{1}{2^{6}}}} + \frac{1}{2} \cdot \sqrt{3} \cdot \sqrt{27} - \sqrt[3]{8}\\& = \sqrt[3]{\sqrt[4]{3^{2} - 2^{8} \cdot \frac{1}{2^{6}}}} + \frac{1}{2} \cdot \sqrt{3} \cdot \sqrt{27} - \sqrt[3]{8}\\& = \sqrt[3]{\sqrt[4]{9 - 2^{8} \cdot \frac{1}{2^{6}}}} + \frac{1}{2} \cdot \sqrt{3} \cdot \sqrt{27} - \sqrt[3]{8}\\& = \sqrt[3]{\sqrt[4]{9 - 256 \cdot \frac{1}{2^{6}}}} + \frac{1}{2} \cdot \sqrt{3} \cdot \sqrt{27} - \sqrt[3]{8}\\& = \sqrt[3]{\sqrt[4]{9 - \frac{256}{64}}} + \frac{1}{2} \cdot \sqrt{3} \cdot \sqrt{27} - \sqrt[3]{8}\\& = \sqrt[3]{\sqrt[4]{9 - 4}} + \frac{1}{2} \cdot \sqrt{3} \cdot \sqrt{27} - \sqrt[3]{8}\\& = \sqrt[3]{\sqrt[4]{5}} + \frac{1}{2} \cdot \sqrt{3} \cdot \sqrt{27} - \sqrt[3]{8}\\& = \sqrt[12]{5} + \frac{1}{2} \cdot \sqrt{3} \cdot \sqrt{27} - \sqrt[3]{8}\\& = \sqrt[12]{5} + \frac{1}{2} \cdot \sqrt{3} \cdot 3 \cdot \sqrt{3} - \sqrt[3]{8}\\& = \sqrt[12]{5} + \frac{9}{2} - \sqrt[3]{8}\\& = \sqrt[12]{5} + \frac{9}{2} - 2\\& = \left(\frac{9}{2} + \sqrt[12]{5}\right) - 2\\& = \frac{5}{2} + \sqrt[12]{5}\end{aligned}\]

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