Solve 21*8/12-sqrt{24}=2x*32

$$21 \times \frac{ 8 }{ 12- \sqrt{ 24 } } =2x \times 32$$

Solve for x

$$x=\frac{7\sqrt{6}}{160}+\frac{21}{80}\approx 0.369665176$$

Steps for Solving Linear Equation

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

$$21\times \frac{8}{12-2\sqrt{6}}=2x\times 32$$

Rationalize the denominator of \frac{8}{12-2\sqrt{6}} by multiplying numerator and denominator by 12+2\sqrt{6}.

$$21\times \frac{8\left(12+2\sqrt{6}\right)}{\left(12-2\sqrt{6}\right)\left(12+2\sqrt{6}\right)}=2x\times 32$$

Consider \left(12-2\sqrt{6}\right)\left(12+2\sqrt{6}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$21\times \frac{8\left(12+2\sqrt{6}\right)}{12^{2}-\left(-2\sqrt{6}\right)^{2}}=2x\times 32$$

Calculate 12 to the power of 2 and get 144.

$$21\times \frac{8\left(12+2\sqrt{6}\right)}{144-\left(-2\sqrt{6}\right)^{2}}=2x\times 32$$

Expand \left(-2\sqrt{6}\right)^{2}.

$$21\times \frac{8\left(12+2\sqrt{6}\right)}{144-\left(-2\right)^{2}\left(\sqrt{6}\right)^{2}}=2x\times 32$$

Calculate -2 to the power of 2 and get 4.

$$21\times \frac{8\left(12+2\sqrt{6}\right)}{144-4\left(\sqrt{6}\right)^{2}}=2x\times 32$$

The square of \sqrt{6} is 6.

$$21\times \frac{8\left(12+2\sqrt{6}\right)}{144-4\times 6}=2x\times 32$$

Multiply 4 and 6 to get 24.

$$21\times \frac{8\left(12+2\sqrt{6}\right)}{144-24}=2x\times 32$$

Subtract 24 from 144 to get 120.

$$21\times \frac{8\left(12+2\sqrt{6}\right)}{120}=2x\times 32$$

Divide 8\left(12+2\sqrt{6}\right) by 120 to get \frac{1}{15}\left(12+2\sqrt{6}\right).

$$21\times \frac{1}{15}\left(12+2\sqrt{6}\right)=2x\times 32$$

Use the distributive property to multiply \frac{1}{15} by 12+2\sqrt{6}.

$$21\left(\frac{1}{15}\times 12+\frac{1}{15}\times 2\sqrt{6}\right)=2x\times 32$$

Multiply \frac{1}{15} and 12 to get \frac{12}{15}.

$$21\left(\frac{12}{15}+\frac{1}{15}\times 2\sqrt{6}\right)=2x\times 32$$

Reduce the fraction \frac{12}{15} to lowest terms by extracting and canceling out 3.

$$21\left(\frac{4}{5}+\frac{1}{15}\times 2\sqrt{6}\right)=2x\times 32$$

Multiply \frac{1}{15} and 2 to get \frac{2}{15}.

$$21\left(\frac{4}{5}+\frac{2}{15}\sqrt{6}\right)=2x\times 32$$

Use the distributive property to multiply 21 by \frac{4}{5}+\frac{2}{15}\sqrt{6}.

$$21\times \frac{4}{5}+21\times \frac{2}{15}\sqrt{6}=2x\times 32$$

Express 21\times \frac{4}{5} as a single fraction.

$$\frac{21\times 4}{5}+21\times \frac{2}{15}\sqrt{6}=2x\times 32$$

Multiply 21 and 4 to get 84.

$$\frac{84}{5}+21\times \frac{2}{15}\sqrt{6}=2x\times 32$$

Express 21\times \frac{2}{15} as a single fraction.

$$\frac{84}{5}+\frac{21\times 2}{15}\sqrt{6}=2x\times 32$$

Multiply 21 and 2 to get 42.

$$\frac{84}{5}+\frac{42}{15}\sqrt{6}=2x\times 32$$

Reduce the fraction \frac{42}{15} to lowest terms by extracting and canceling out 3.

$$\frac{84}{5}+\frac{14}{5}\sqrt{6}=2x\times 32$$

Multiply 2 and 32 to get 64.

$$\frac{84}{5}+\frac{14}{5}\sqrt{6}=64x$$

Swap sides so that all variable terms are on the left hand side.

$$64x=\frac{84}{5}+\frac{14}{5}\sqrt{6}$$

The equation is in standard form.

$$64x=\frac{14\sqrt{6}+84}{5}$$

Divide both sides by 64.

$$\frac{64x}{64}=\frac{14\sqrt{6}+84}{5\times 64}$$

Dividing by 64 undoes the multiplication by 64.

$$x=\frac{14\sqrt{6}+84}{5\times 64}$$

Divide \frac{84+14\sqrt{6}}{5} by 64.

$$x=\frac{7\sqrt{6}}{160}+\frac{21}{80}$$