square root of x^4 +64 - 16^2

asked by guest
on Nov 23, 2024 at 1:03 am



You asked:

Evaluate the expression: \(\sqrt{{x}^{4} + 64 - {16}^{2}}\)

MathBot Answer:

Evaluated



\(\displaystyle \sqrt{{x}^{4} + 64 - {16}^{2}} = \sqrt{x^{4} - 192} \)

Expanded

\[\sqrt{{x}^{4} + 64 - {16}^{2}} = \sqrt{x^{4} - 192}\]

Roots

\[\sqrt{{x}^{4} + 64 - {16}^{2}} = i \sqrt[4]{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} - 192\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3},\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} - 192 \right)}}{2} \right)} + \sqrt[4]{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} - 192\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3},\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} - 192 \right)}}{2} \right)} \approx 13.856406460551 i \left(0.000434027777777778 \left(\left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3}\right)^{2} + \left(0.00520833333333333 \left(\operatorname{re}{\left(x\right)}\right)^{4} - 0.03125 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + 0.00520833333333333 \left(\operatorname{im}{\left(x\right)}\right)^{4} - 1\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3},\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} - 192 \right)}}{2} \right)} + 13.856406460551 \left(0.000434027777777778 \left(\left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3}\right)^{2} + \left(0.00520833333333333 \left(\operatorname{re}{\left(x\right)}\right)^{4} - 0.03125 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + 0.00520833333333333 \left(\operatorname{im}{\left(x\right)}\right)^{4} - 1\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3},\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} - 192 \right)}}{2} \right)}\]\[\sqrt{{x}^{4} + 64 - {16}^{2}} = i \left(- \sqrt[4]{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} - 192\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3},\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} - 192 \right)}}{2} \right)}\right) - \sqrt[4]{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} - 192\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3},\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} - 192 \right)}}{2} \right)} \approx - 13.856406460551 i \left(0.000434027777777778 \left(\left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3}\right)^{2} + \left(0.00520833333333333 \left(\operatorname{re}{\left(x\right)}\right)^{4} - 0.03125 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + 0.00520833333333333 \left(\operatorname{im}{\left(x\right)}\right)^{4} - 1\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3},\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} - 192 \right)}}{2} \right)} - 13.856406460551 \left(0.000434027777777778 \left(\left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3}\right)^{2} + \left(0.00520833333333333 \left(\operatorname{re}{\left(x\right)}\right)^{4} - 0.03125 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + 0.00520833333333333 \left(\operatorname{im}{\left(x\right)}\right)^{4} - 1\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3},\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} - 192 \right)}}{2} \right)}\]

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