Solve (frac{sqrt{24sqrt{2}}3}{4}+3sqrt{2}-7/4)K-K=frac{sqrt{8sqrt{2}-1}}{4}-sqrt{6}-frac{sqrt{3}}{4}

$$( \frac { \sqrt { 24 \sqrt { 2 } } 3 } { 4 } + 3 \sqrt { 2 } - \frac { 7 } { 4 } ) K - K = \frac { \sqrt { 8 \sqrt { 2 } - 1 } } { 4 } - \sqrt { 6 } - \frac { \sqrt { 3 } } { 4 }$$

Solve for K

$$K=-\frac{-\sqrt{8\sqrt{2}-1}+\sqrt{3}+4\sqrt{6}}{6\sqrt[4]{72}+12\sqrt{2}-11}\approx -0.354760275$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 4.

$$4\left(\frac{3\sqrt{24\sqrt{2}}}{4}+3\sqrt{2}-\frac{7}{4}\right)K-4K=\sqrt{8\sqrt{2}-1}-4\sqrt{6}-\sqrt{3}$$

Since \frac{3\sqrt{24\sqrt{2}}}{4} and \frac{7}{4} have the same denominator, subtract them by subtracting their numerators.

$$4\left(\frac{3\sqrt{24\sqrt{2}}-7}{4}+3\sqrt{2}\right)K-4K=\sqrt{8\sqrt{2}-1}-4\sqrt{6}-\sqrt{3}$$

Do the multiplications in 3\sqrt{24\sqrt{2}}-7.

$$4\left(\frac{6\sqrt[4]{72}-7}{4}+3\sqrt{2}\right)K-4K=\sqrt{8\sqrt{2}-1}-4\sqrt{6}-\sqrt{3}$$

Use the distributive property to multiply 4 by \frac{6\sqrt[4]{72}-7}{4}+3\sqrt{2}.

$$\left(4\times \frac{6\sqrt[4]{72}-7}{4}+12\sqrt{2}\right)K-4K=\sqrt{8\sqrt{2}-1}-4\sqrt{6}-\sqrt{3}$$

Express 4\times \frac{6\sqrt[4]{72}-7}{4} as a single fraction.

$$\left(\frac{4\left(6\sqrt[4]{72}-7\right)}{4}+12\sqrt{2}\right)K-4K=\sqrt{8\sqrt{2}-1}-4\sqrt{6}-\sqrt{3}$$

Cancel out 4 and 4.

$$\left(6\sqrt[4]{72}-7+12\sqrt{2}\right)K-4K=\sqrt{8\sqrt{2}-1}-4\sqrt{6}-\sqrt{3}$$

Use the distributive property to multiply 6\sqrt[4]{72}-7+12\sqrt{2} by K.

$$6\sqrt[4]{72}K-7K+12\sqrt{2}K-4K=\sqrt{8\sqrt{2}-1}-4\sqrt{6}-\sqrt{3}$$

Combine -7K and -4K to get -11K.

$$6\sqrt[4]{72}K-11K+12\sqrt{2}K=\sqrt{8\sqrt{2}-1}-4\sqrt{6}-\sqrt{3}$$

Combine all terms containing K.

$$\left(6\sqrt[4]{72}-11+12\sqrt{2}\right)K=\sqrt{8\sqrt{2}-1}-4\sqrt{6}-\sqrt{3}$$

The equation is in standard form.

$$\left(6\sqrt[4]{72}+12\sqrt{2}-11\right)K=\sqrt{8\sqrt{2}-1}-\sqrt{3}-4\sqrt{6}$$

Divide both sides by 6\sqrt[4]{72}-11+12\sqrt{2}.

$$\frac{\left(6\sqrt[4]{72}+12\sqrt{2}-11\right)K}{6\sqrt[4]{72}+12\sqrt{2}-11}=\frac{\sqrt{8\sqrt{2}-1}-\sqrt{3}-4\sqrt{6}}{6\sqrt[4]{72}+12\sqrt{2}-11}$$

Dividing by 6\sqrt[4]{72}-11+12\sqrt{2} undoes the multiplication by 6\sqrt[4]{72}-11+12\sqrt{2}.

$$K=\frac{\sqrt{8\sqrt{2}-1}-\sqrt{3}-4\sqrt{6}}{6\sqrt[4]{72}+12\sqrt{2}-11}$$