Solve t(-2/3)^k-2X(-2/3)^3=(-2/3)^-2

$$t ( \frac { - 2 } { 3 } ) ^ { k - 2 } X ( \frac { - 2 } { 3 } ) ^ { 3 } = ( \frac { - 2 } { 3 } ) ^ { - 2 }$$

Solve for X (complex solution)

$$X=-\frac{27}{8t\left(-\frac{2}{3}\right)^{k}}$$

$$t\neq 0$$

Steps for Solving Linear Equation

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

$$t\left(-\frac{2}{3}\right)^{k-2}X\times \left(\frac{-2}{3}\right)^{3}=\left(\frac{-2}{3}\right)^{-2}$$

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

$$t\left(-\frac{2}{3}\right)^{k-2}X\left(-\frac{2}{3}\right)^{3}=\left(\frac{-2}{3}\right)^{-2}$$

Calculate -\frac{2}{3} to the power of 3 and get -\frac{8}{27}.

$$t\left(-\frac{2}{3}\right)^{k-2}X\left(-\frac{8}{27}\right)=\left(\frac{-2}{3}\right)^{-2}$$

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

$$t\left(-\frac{2}{3}\right)^{k-2}X\left(-\frac{8}{27}\right)=\left(-\frac{2}{3}\right)^{-2}$$

Calculate -\frac{2}{3} to the power of -2 and get \frac{9}{4}.

$$t\left(-\frac{2}{3}\right)^{k-2}X\left(-\frac{8}{27}\right)=\frac{9}{4}$$

The equation is in standard form.

$$\left(-\frac{8t\left(-\frac{2}{3}\right)^{k-2}}{27}\right)X=\frac{9}{4}$$

Divide both sides by -\frac{8}{27}t\left(-\frac{2}{3}\right)^{k-2}.

$$\frac{\left(-\frac{8t\left(-\frac{2}{3}\right)^{k-2}}{27}\right)X}{-\frac{8t\left(-\frac{2}{3}\right)^{k-2}}{27}}=\frac{\frac{9}{4}}{-\frac{8t\left(-\frac{2}{3}\right)^{k-2}}{27}}$$

Dividing by -\frac{8}{27}t\left(-\frac{2}{3}\right)^{k-2} undoes the multiplication by -\frac{8}{27}t\left(-\frac{2}{3}\right)^{k-2}.

$$X=\frac{\frac{9}{4}}{-\frac{8t\left(-\frac{2}{3}\right)^{k-2}}{27}}$$

Divide \frac{9}{4} by -\frac{8}{27}t\left(-\frac{2}{3}\right)^{k-2}.

$$X=-\frac{27}{8t\left(-\frac{2}{3}\right)^{k}}$$

Solve for X

$$X=-\frac{27}{8t\left(-\frac{2}{3}\right)^{k}}$$

$$t\neq 0\text{ and }\left(-\frac{2}{3}\right)^{k+1}\neq 0\text{ and }Denominator(k)\text{bmod}2=1$$

Steps for Solving Linear Equation

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

$$t\left(-\frac{2}{3}\right)^{k-2}X\times \left(\frac{-2}{3}\right)^{3}=\left(\frac{-2}{3}\right)^{-2}$$

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

$$t\left(-\frac{2}{3}\right)^{k-2}X\left(-\frac{2}{3}\right)^{3}=\left(\frac{-2}{3}\right)^{-2}$$

Calculate -\frac{2}{3} to the power of 3 and get -\frac{8}{27}.

$$t\left(-\frac{2}{3}\right)^{k-2}X\left(-\frac{8}{27}\right)=\left(\frac{-2}{3}\right)^{-2}$$

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

$$t\left(-\frac{2}{3}\right)^{k-2}X\left(-\frac{8}{27}\right)=\left(-\frac{2}{3}\right)^{-2}$$

Calculate -\frac{2}{3} to the power of -2 and get \frac{9}{4}.

$$t\left(-\frac{2}{3}\right)^{k-2}X\left(-\frac{8}{27}\right)=\frac{9}{4}$$

The equation is in standard form.

$$\left(-\frac{8t\left(-\frac{2}{3}\right)^{k-2}}{27}\right)X=\frac{9}{4}$$

Divide both sides by -\frac{8}{27}t\left(-\frac{2}{3}\right)^{k-2}.

$$\frac{\left(-\frac{8t\left(-\frac{2}{3}\right)^{k-2}}{27}\right)X}{-\frac{8t\left(-\frac{2}{3}\right)^{k-2}}{27}}=\frac{\frac{9}{4}}{-\frac{8t\left(-\frac{2}{3}\right)^{k-2}}{27}}$$

Dividing by -\frac{8}{27}t\left(-\frac{2}{3}\right)^{k-2} undoes the multiplication by -\frac{8}{27}t\left(-\frac{2}{3}\right)^{k-2}.

$$X=\frac{\frac{9}{4}}{-\frac{8t\left(-\frac{2}{3}\right)^{k-2}}{27}}$$

Divide \frac{9}{4} by -\frac{8}{27}t\left(-\frac{2}{3}\right)^{k-2}.

$$X=-\frac{27}{8t\left(-\frac{2}{3}\right)^{k}}$$

Solve for k (complex solution)

$$k=\frac{\ln(-\frac{243}{32Xt})-\ln(\frac{9}{4})+2\pi i}{\ln(\frac{2}{3})+\pi i}+\frac{2\pi n_{1}i}{\ln(\frac{2}{3})+\pi i}$$

$$n_{1}\in \mathrm{Z}$$

$$Xt\neq 0$$

Solve for k

$$k=-\frac{-\ln(-\frac{243}{32Xt})+\ln(\frac{9}{4})}{\ln(\frac{2}{3})}$$

$$Xt\neq 0\text{ and }\left(t>0\text{ or }X>0\right)\text{ and }\left(X<0\text{ or }t<0\right)\text{ and }Numerator(\frac{\ln(-\frac{243}{32Xt})}{\ln(\frac{2}{3})})\text{bmod}2=0\text{ and }Denominator(\frac{\ln(-\frac{243}{32Xt})}{\ln(\frac{2}{3})})\text{bmod}2=1\text{ and }Denominator(\frac{-\ln(-\frac{243}{32Xt})+\ln(\frac{9}{4})}{\ln(\frac{2}{3})})\text{bmod}2=1$$