x⁻² ÷ (x² ÷ x⁻⁶ • 4x²) = 33

asked by guest
on Apr 07, 2025 at 12:32 am



You asked:

Solve the equation x−2x2x−6⋅4x2=33\frac{{x}^{-2}}{\frac{{x}^{2}}{{x}^{-6}} \cdot 4 {x}^{2}} = 33 for the variable xx.

MathBot Answer:

The solutions are given by the formula: x=256⋅331112cos⁡(πk6)66+256⋅331112sin⁡(πk6)66ix = \frac{2^{\frac{5}{6}} \cdot 33^{\frac{11}{12}} \cos{\left(\frac{\pi k}{6} \right)}}{66} + \frac{2^{\frac{5}{6}} \cdot 33^{\frac{11}{12}} \sin{\left(\frac{\pi k}{6} \right)}}{66} i where kk is an integer in 0≤k<120 \le k < 12.

There are 22 real solutions.

There are 1010 solutions with nonzero imaginary part.

Real solutions: x=256⋅33111266≈0.66571065x=−256⋅33111266≈−0.66571065\begin{aligned}x &= \frac{2^{\frac{5}{6}} \cdot 33^{\frac{11}{12}}}{66} \\&\approx 0.66571065\\x &= - \frac{2^{\frac{5}{6}} \cdot 33^{\frac{11}{12}}}{66} \\&\approx -0.66571065\end{aligned}

Solutions with nonzero imaginary part (88 of 1010 displayed): x=111112⋅256⋅351244+256⋅331112i132≈0.57652234+0.33285533ix=256⋅331112132+111112⋅256⋅3512i44≈0.33285533+0.57652234ix=256⋅33111266i≈0.66571065ix=−256⋅331112132+111112⋅256⋅3512i44≈−0.33285533+0.57652234ix=−111112⋅256⋅351244+256⋅331112i132≈−0.57652234+0.33285533ix=−111112⋅256⋅351244−256⋅331112i132≈−0.57652234−0.33285533ix=−256⋅331112132−111112⋅256⋅3512i44≈−0.33285533−0.57652234ix=−256⋅33111266i≈−0.66571065i\begin{aligned}x &= \frac{11^{\frac{11}{12}} \cdot 2^{\frac{5}{6}} \cdot 3^{\frac{5}{12}}}{44} + \frac{2^{\frac{5}{6}} \cdot 33^{\frac{11}{12}} i}{132} \\&\approx 0.57652234 + 0.33285533 i\\x &= \frac{2^{\frac{5}{6}} \cdot 33^{\frac{11}{12}}}{132} + \frac{11^{\frac{11}{12}} \cdot 2^{\frac{5}{6}} \cdot 3^{\frac{5}{12}} i}{44} \\&\approx 0.33285533 + 0.57652234 i\\x &= \frac{2^{\frac{5}{6}} \cdot 33^{\frac{11}{12}}}{66} i \\&\approx 0.66571065 i\\x &= - \frac{2^{\frac{5}{6}} \cdot 33^{\frac{11}{12}}}{132} + \frac{11^{\frac{11}{12}} \cdot 2^{\frac{5}{6}} \cdot 3^{\frac{5}{12}} i}{44} \\&\approx -0.33285533 + 0.57652234 i\\x &= - \frac{11^{\frac{11}{12}} \cdot 2^{\frac{5}{6}} \cdot 3^{\frac{5}{12}}}{44} + \frac{2^{\frac{5}{6}} \cdot 33^{\frac{11}{12}} i}{132} \\&\approx -0.57652234 + 0.33285533 i\\x &= - \frac{11^{\frac{11}{12}} \cdot 2^{\frac{5}{6}} \cdot 3^{\frac{5}{12}}}{44} - \frac{2^{\frac{5}{6}} \cdot 33^{\frac{11}{12}} i}{132} \\&\approx -0.57652234 -0.33285533 i\\x &= - \frac{2^{\frac{5}{6}} \cdot 33^{\frac{11}{12}}}{132} - \frac{11^{\frac{11}{12}} \cdot 2^{\frac{5}{6}} \cdot 3^{\frac{5}{12}} i}{44} \\&\approx -0.33285533 -0.57652234 i\\x &= - \frac{2^{\frac{5}{6}} \cdot 33^{\frac{11}{12}}}{66} i \\&\approx -0.66571065 i\end{aligned}

ii is the imaginary unit, defined as i2=−1i^2 = -1.

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