Solve (2/k^2)^2+(2/k)^2=36

$$( \frac { 2 } { k ^ { 2 } } ) ^ { 2 } + ( \frac { 2 } { k } ) ^ { 2 } = 36$$

Solve for k

$$k=\frac{\sqrt{2\left(\sqrt{37}+1\right)}}{6}\approx 0.627285268$$

$$k=-\frac{\sqrt{2\left(\sqrt{37}+1\right)}}{6}\approx -0.627285268$$

Steps Using the Quadratic Formula

To raise \frac{2}{k^{2}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{2^{2}}{\left(k^{2}\right)^{2}}+\left(\frac{2}{k}\right)^{2}=36$$

To raise \frac{2}{k} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{2^{2}}{\left(k^{2}\right)^{2}}+\frac{2^{2}}{k^{2}}=36$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(k^{2}\right)^{2} and k^{2} is k^{4}. Multiply \frac{2^{2}}{k^{2}} times \frac{k^{2}}{k^{2}}.

$$\frac{2^{2}}{k^{4}}+\frac{2^{2}k^{2}}{k^{4}}=36$$

Since \frac{2^{2}}{k^{4}} and \frac{2^{2}k^{2}}{k^{4}} have the same denominator, add them by adding their numerators.

$$\frac{2^{2}+2^{2}k^{2}}{k^{4}}=36$$

Do the multiplications in 2^{2}+2^{2}k^{2}.

$$\frac{2^{2}+4k^{2}}{k^{4}}=36$$

Combine like terms in 2^{2}+4k^{2}.

$$\frac{4+4k^{2}}{k^{4}}=36$$

Subtract 36 from both sides.

$$\frac{4+4k^{2}}{k^{4}}-36=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 36 times \frac{k^{4}}{k^{4}}.

$$\frac{4+4k^{2}}{k^{4}}-\frac{36k^{4}}{k^{4}}=0$$

Since \frac{4+4k^{2}}{k^{4}} and \frac{36k^{4}}{k^{4}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4+4k^{2}-36k^{4}}{k^{4}}=0$$

Variable k cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by k^{4}.

$$4+4k^{2}-36k^{4}=0$$

Substitute t for k^{2}.

$$-36t^{2}+4t+4=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute -36 for a, 4 for b, and 4 for c in the quadratic formula.

$$t=\frac{-4±\sqrt{4^{2}-4\left(-36\right)\times 4}}{-36\times 2}$$

Do the calculations.

$$t=\frac{-4±4\sqrt{37}}{-72}$$

Solve the equation t=\frac{-4±4\sqrt{37}}{-72} when ± is plus and when ± is minus.

$$t=\frac{1-\sqrt{37}}{18} t=\frac{\sqrt{37}+1}{18}$$

Since k=t^{2}, the solutions are obtained by evaluating k=±\sqrt{t} for positive t.

$$k=\frac{\sqrt{2\sqrt{37}+2}}{6} k=-\frac{\sqrt{2\sqrt{37}+2}}{6}$$