\sqrt((x+4)/(x-3))

asked by guest
on Nov 23, 2024 at 5:56 pm



You asked:

Evaluate the expression: \(\sqrt{\frac{x + 4}{x - 3}}\)

MathBot Answer:

Evaluated



\(\displaystyle \sqrt{\frac{x + 4}{x - 3}} = \sqrt{\frac{x + 4}{x - 3}} \)

Expanded

\[\sqrt{\frac{x + 4}{x - 3}} = \sqrt{\frac{x}{x - 3} + \frac{4}{x - 3}}\]

Roots

\[\sqrt{\frac{x + 4}{x - 3}} = i \sqrt[4]{\left(- \frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \left(\operatorname{re}{\left(x\right)} - 3\right)}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}\right)^{2} + \left(\frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{re}{\left(x\right)} - 3\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{re}{\left(x\right)} - 3\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}},- \frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \left(\operatorname{re}{\left(x\right)} - 3\right)}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)} + \sqrt[4]{\left(- \frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \left(\operatorname{re}{\left(x\right)} - 3\right)}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}\right)^{2} + \left(\frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{re}{\left(x\right)} - 3\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{re}{\left(x\right)} - 3\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}},- \frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \left(\operatorname{re}{\left(x\right)} - 3\right)}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)} \approx i \left(\left(- \frac{\left(- \operatorname{re}{\left(x\right)} - 4.0\right) \left(\operatorname{re}{\left(x\right)} - 3.0\right)}{9.0 \left(1 - 0.333333333333333 \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{9.0 \left(1 - 0.333333333333333 \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}\right)^{2} + \left(\frac{\left(- \operatorname{re}{\left(x\right)} - 4.0\right) \operatorname{im}{\left(x\right)}}{9.0 \left(1 - 0.333333333333333 \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{re}{\left(x\right)} - 3.0\right) \operatorname{im}{\left(x\right)}}{9.0 \left(1 - 0.333333333333333 \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{re}{\left(x\right)} - 3\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}},- \frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \left(\operatorname{re}{\left(x\right)} - 3\right)}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)} + \left(\left(- \frac{\left(- \operatorname{re}{\left(x\right)} - 4.0\right) \left(\operatorname{re}{\left(x\right)} - 3.0\right)}{9.0 \left(1 - 0.333333333333333 \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{9.0 \left(1 - 0.333333333333333 \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}\right)^{2} + \left(\frac{\left(- \operatorname{re}{\left(x\right)} - 4.0\right) \operatorname{im}{\left(x\right)}}{9.0 \left(1 - 0.333333333333333 \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{re}{\left(x\right)} - 3.0\right) \operatorname{im}{\left(x\right)}}{9.0 \left(1 - 0.333333333333333 \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{re}{\left(x\right)} - 3\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}},- \frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \left(\operatorname{re}{\left(x\right)} - 3\right)}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)}\]\[\sqrt{\frac{x + 4}{x - 3}} = i \left(- \sqrt[4]{\left(- \frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \left(\operatorname{re}{\left(x\right)} - 3\right)}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}\right)^{2} + \left(\frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{re}{\left(x\right)} - 3\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{re}{\left(x\right)} - 3\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}},- \frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \left(\operatorname{re}{\left(x\right)} - 3\right)}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)}\right) - \sqrt[4]{\left(- \frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \left(\operatorname{re}{\left(x\right)} - 3\right)}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}\right)^{2} + \left(\frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{re}{\left(x\right)} - 3\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{re}{\left(x\right)} - 3\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}},- \frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \left(\operatorname{re}{\left(x\right)} - 3\right)}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)} \approx - i \left(\left(- \frac{\left(- \operatorname{re}{\left(x\right)} - 4.0\right) \left(\operatorname{re}{\left(x\right)} - 3.0\right)}{9.0 \left(1 - 0.333333333333333 \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{9.0 \left(1 - 0.333333333333333 \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}\right)^{2} + \left(\frac{\left(- \operatorname{re}{\left(x\right)} - 4.0\right) \operatorname{im}{\left(x\right)}}{9.0 \left(1 - 0.333333333333333 \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{re}{\left(x\right)} - 3.0\right) \operatorname{im}{\left(x\right)}}{9.0 \left(1 - 0.333333333333333 \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{re}{\left(x\right)} - 3\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}},- \frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \left(\operatorname{re}{\left(x\right)} - 3\right)}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)} - \left(\left(- \frac{\left(- \operatorname{re}{\left(x\right)} - 4.0\right) \left(\operatorname{re}{\left(x\right)} - 3.0\right)}{9.0 \left(1 - 0.333333333333333 \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{9.0 \left(1 - 0.333333333333333 \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}\right)^{2} + \left(\frac{\left(- \operatorname{re}{\left(x\right)} - 4.0\right) \operatorname{im}{\left(x\right)}}{9.0 \left(1 - 0.333333333333333 \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{re}{\left(x\right)} - 3.0\right) \operatorname{im}{\left(x\right)}}{9.0 \left(1 - 0.333333333333333 \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{re}{\left(x\right)} - 3\right) \operatorname{im}{\left(x\right)}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}},- \frac{\left(- \operatorname{re}{\left(x\right)} - 4\right) \left(\operatorname{re}{\left(x\right)} - 3\right)}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(3 - \operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)}\]

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