Solve n/3+n=sqrt{3div8}

$$\frac{ n }{ 3+n } = \sqrt{ 3 \div 8 }$$

Solve for n

$$n = \frac{6 \sqrt{6} + 9}{5} \approx 4.739387691$$

Steps for Solving Linear Equation

Variable n cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by n+3.

$$n=\left(n+3\right)\sqrt{\frac{3}{8}}$$

Rewrite the square root of the division \sqrt{\frac{3}{8}} as the division of square roots \frac{\sqrt{3}}{\sqrt{8}}.

$$n=\left(n+3\right)\times \frac{\sqrt{3}}{\sqrt{8}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$n=\left(n+3\right)\times \frac{\sqrt{3}}{2\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{3}}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$n=\left(n+3\right)\times \frac{\sqrt{3}\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$n=\left(n+3\right)\times \frac{\sqrt{3}\sqrt{2}}{2\times 2}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$n=\left(n+3\right)\times \frac{\sqrt{6}}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$n=\left(n+3\right)\times \frac{\sqrt{6}}{4}$$

Express \left(n+3\right)\times \frac{\sqrt{6}}{4} as a single fraction.

$$n=\frac{\left(n+3\right)\sqrt{6}}{4}$$

Use the distributive property to multiply n+3 by \sqrt{6}.

$$n=\frac{n\sqrt{6}+3\sqrt{6}}{4}$$

Subtract \frac{n\sqrt{6}+3\sqrt{6}}{4} from both sides.

$$n-\frac{n\sqrt{6}+3\sqrt{6}}{4}=0$$

Multiply both sides of the equation by 4.

$$4n-\left(n\sqrt{6}+3\sqrt{6}\right)=0$$

To find the opposite of n\sqrt{6}+3\sqrt{6}, find the opposite of each term.

$$4n-n\sqrt{6}-3\sqrt{6}=0$$

Add 3\sqrt{6} to both sides. Anything plus zero gives itself.

$$4n-n\sqrt{6}=3\sqrt{6}$$

Combine all terms containing n.

$$\left(4-\sqrt{6}\right)n=3\sqrt{6}$$

Divide both sides by 4-\sqrt{6}.

$$\frac{\left(4-\sqrt{6}\right)n}{4-\sqrt{6}}=\frac{3\sqrt{6}}{4-\sqrt{6}}$$

Dividing by 4-\sqrt{6} undoes the multiplication by 4-\sqrt{6}.

$$n=\frac{3\sqrt{6}}{4-\sqrt{6}}$$

Divide 3\sqrt{6} by 4-\sqrt{6}.

$$n=\frac{6\sqrt{6}+9}{5}$$