\left(\frac{\left(\sqrt{2^2+4}\right)^5}{5}-\frac{4\left(\sqrt{2^2+4}\right)^3}{3}\ \right)-\left(\frac{\left(\sqrt{0^2+4}\right)^5}{5}-\frac{4\left(\sqrt{0^2+4}\right)^3}{3}\right)

asked by guest
on Oct 25, 2024 at 9:25 am



You asked:

Evaluate the expression: \(\frac{{\sqrt{{2}^{2} + 4}}^{5}}{5} - \frac{4 \cdot {\sqrt{{2}^{2} + 4}}^{3}}{3} - \left(\frac{{\sqrt{{0}^{2} + 4}}^{5}}{5} - \frac{4 \cdot {\sqrt{{0}^{2} + 4}}^{3}}{3}\right)\)

MathBot Answer:

\[\frac{{\sqrt{{2}^{2} + 4}}^{5}}{5} - \frac{4 \cdot {\sqrt{{2}^{2} + 4}}^{3}}{3} - \left(\frac{{\sqrt{{0}^{2} + 4}}^{5}}{5} - \frac{4 \cdot {\sqrt{{0}^{2} + 4}}^{3}}{3}\right) = \frac{64}{15} + \frac{64 \sqrt{2}}{15} \approx 10.30064453279187220822053855662805\]

\[\begin{aligned}\frac{{\sqrt{{2}^{2} + 4}}^{5}}{5} - \frac{4 \cdot {\sqrt{{2}^{2} + 4}}^{3}}{3} - \left(\frac{{\sqrt{{0}^{2} + 4}}^{5}}{5} - \frac{4 \cdot {\sqrt{{0}^{2} + 4}}^{3}}{3}\right)& = \left(\sqrt{2^{2} + 4}\right)^{5} \cdot \frac{1}{5} - 4 \cdot \left(\sqrt{2^{2} + 4}\right)^{3} \cdot \frac{1}{3} - \left(\left(\sqrt{0^{2} + 4}\right)^{5} \cdot \frac{1}{5} - 4 \cdot \left(\sqrt{0^{2} + 4}\right)^{3} \cdot \frac{1}{3}\right)\\& = \left(\sqrt{4 + 4}\right)^{5} \cdot \frac{1}{5} - 4 \cdot \left(\sqrt{2^{2} + 4}\right)^{3} \cdot \frac{1}{3} - \left(\left(\sqrt{0^{2} + 4}\right)^{5} \cdot \frac{1}{5} - 4 \cdot \left(\sqrt{0^{2} + 4}\right)^{3} \cdot \frac{1}{3}\right)\\& = \left(\sqrt{8}\right)^{5} \cdot \frac{1}{5} - 4 \cdot \left(\sqrt{2^{2} + 4}\right)^{3} \cdot \frac{1}{3} - \left(\left(\sqrt{0^{2} + 4}\right)^{5} \cdot \frac{1}{5} - 4 \cdot \left(\sqrt{0^{2} + 4}\right)^{3} \cdot \frac{1}{3}\right)\\& = \left(2 \cdot \sqrt{2}\right)^{5} \cdot \frac{1}{5} - 4 \cdot \left(\sqrt{2^{2} + 4}\right)^{3} \cdot \frac{1}{3} - \left(\left(\sqrt{0^{2} + 4}\right)^{5} \cdot \frac{1}{5} - 4 \cdot \left(\sqrt{0^{2} + 4}\right)^{3} \cdot \frac{1}{3}\right)\\& = 128 \cdot \sqrt{2} \cdot \frac{1}{5} - 4 \cdot \left(\sqrt{2^{2} + 4}\right)^{3} \cdot \frac{1}{3} - \left(\left(\sqrt{0^{2} + 4}\right)^{5} \cdot \frac{1}{5} - 4 \cdot \left(\sqrt{0^{2} + 4}\right)^{3} \cdot \frac{1}{3}\right)\\& = \frac{128}{5} \cdot \sqrt{2} - 4 \cdot \left(\sqrt{2^{2} + 4}\right)^{3} \cdot \frac{1}{3} - \left(\left(\sqrt{0^{2} + 4}\right)^{5} \cdot \frac{1}{5} - 4 \cdot \left(\sqrt{0^{2} + 4}\right)^{3} \cdot \frac{1}{3}\right)\\& = \frac{128}{5} \cdot \sqrt{2} - 4 \cdot \left(\sqrt{4 + 4}\right)^{3} \cdot \frac{1}{3} - \left(\left(\sqrt{0^{2} + 4}\right)^{5} \cdot \frac{1}{5} - 4 \cdot \left(\sqrt{0^{2} + 4}\right)^{3} \cdot \frac{1}{3}\right)\\& = \frac{128}{5} \cdot \sqrt{2} - 4 \cdot \left(\sqrt{8}\right)^{3} \cdot \frac{1}{3} - \left(\left(\sqrt{0^{2} + 4}\right)^{5} \cdot \frac{1}{5} - 4 \cdot \left(\sqrt{0^{2} + 4}\right)^{3} \cdot \frac{1}{3}\right)\\& = \frac{128}{5} \cdot \sqrt{2} - 4 \cdot \left(2 \cdot \sqrt{2}\right)^{3} \cdot \frac{1}{3} - \left(\left(\sqrt{0^{2} + 4}\right)^{5} \cdot \frac{1}{5} - 4 \cdot \left(\sqrt{0^{2} + 4}\right)^{3} \cdot \frac{1}{3}\right)\\& = \frac{128}{5} \cdot \sqrt{2} - 4 \cdot 16 \cdot \sqrt{2} \cdot \frac{1}{3} - \left(\left(\sqrt{0^{2} + 4}\right)^{5} \cdot \frac{1}{5} - 4 \cdot \left(\sqrt{0^{2} + 4}\right)^{3} \cdot \frac{1}{3}\right)\\& = \frac{128}{5} \cdot \sqrt{2} - 4 \cdot \frac{16}{3} \cdot \sqrt{2} - \left(\left(\sqrt{0^{2} + 4}\right)^{5} \cdot \frac{1}{5} - 4 \cdot \left(\sqrt{0^{2} + 4}\right)^{3} \cdot \frac{1}{3}\right)\\& = \frac{128}{5} \cdot \sqrt{2} - \frac{64}{3} \cdot \sqrt{2} - \left(\left(\sqrt{0^{2} + 4}\right)^{5} \cdot \frac{1}{5} - 4 \cdot \left(\sqrt{0^{2} + 4}\right)^{3} \cdot \frac{1}{3}\right)\\& = \frac{128}{5} \cdot \sqrt{2} - \frac{64}{3} \cdot \sqrt{2} - \left(\left(\sqrt{0 + 4}\right)^{5} \cdot \frac{1}{5} - 4 \cdot \left(\sqrt{0^{2} + 4}\right)^{3} \cdot \frac{1}{3}\right)\\& = \frac{128}{5} \cdot \sqrt{2} - \frac{64}{3} \cdot \sqrt{2} - \left(\left(\sqrt{4}\right)^{5} \cdot \frac{1}{5} - 4 \cdot \left(\sqrt{0^{2} + 4}\right)^{3} \cdot \frac{1}{3}\right)\\& = \frac{128}{5} \cdot \sqrt{2} - \frac{64}{3} \cdot \sqrt{2} - \left(2^{5} \cdot \frac{1}{5} - 4 \cdot \left(\sqrt{0^{2} + 4}\right)^{3} \cdot \frac{1}{3}\right)\\& = \frac{128}{5} \cdot \sqrt{2} - \frac{64}{3} \cdot \sqrt{2} - \left(\frac{32}{5} - 4 \cdot \left(\sqrt{0^{2} + 4}\right)^{3} \cdot \frac{1}{3}\right)\\& = \frac{128}{5} \cdot \sqrt{2} - \frac{64}{3} \cdot \sqrt{2} - \left(\frac{32}{5} - 4 \cdot \left(\sqrt{0 + 4}\right)^{3} \cdot \frac{1}{3}\right)\\& = \frac{128}{5} \cdot \sqrt{2} - \frac{64}{3} \cdot \sqrt{2} - \left(\frac{32}{5} - 4 \cdot \left(\sqrt{4}\right)^{3} \cdot \frac{1}{3}\right)\\& = \frac{128}{5} \cdot \sqrt{2} - \frac{64}{3} \cdot \sqrt{2} - \left(\frac{32}{5} - 4 \cdot 2^{3} \cdot \frac{1}{3}\right)\\& = \frac{128}{5} \cdot \sqrt{2} - \frac{64}{3} \cdot \sqrt{2} - \left(\frac{32}{5} - 4 \cdot \frac{8}{3}\right)\\& = \frac{128}{5} \cdot \sqrt{2} - \frac{64}{3} \cdot \sqrt{2} - \left(\frac{32}{5} - \frac{32}{3}\right)\\& = \frac{128}{5} \cdot \sqrt{2} - \frac{64}{3} \cdot \sqrt{2} - - \frac{64}{15}\\& = \frac{128}{5} \cdot \sqrt{2} - \frac{64}{3} \cdot \sqrt{2} + \frac{64}{15}\\& = \frac{64}{15} \cdot \sqrt{2} + \frac{64}{15}\\& = \frac{64}{15} + \frac{64}{15} \cdot \sqrt{2}\end{aligned}\]

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