Solve x=frac{4sqrt{3}}{3-sqrt{2}}

$$x = \frac { 4 \sqrt { 3 } } { 3 - \sqrt { 2 } }$$

Solve for x

$$x = \frac{4 {(\sqrt{6} + 3 \sqrt{3})}}{7} \approx 4.36893838$$

Steps for Solving Linear Equation

Rationalize the denominator of \frac{4\sqrt{3}}{3-\sqrt{2}} by multiplying numerator and denominator by 3+\sqrt{2}.

$$x=\frac{4\sqrt{3}\left(3+\sqrt{2}\right)}{\left(3-\sqrt{2}\right)\left(3+\sqrt{2}\right)}$$

Consider \left(3-\sqrt{2}\right)\left(3+\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$x=\frac{4\sqrt{3}\left(3+\sqrt{2}\right)}{3^{2}-\left(\sqrt{2}\right)^{2}}$$

Square 3. Square \sqrt{2}.

$$x=\frac{4\sqrt{3}\left(3+\sqrt{2}\right)}{9-2}$$

Subtract 2 from 9 to get 7.

$$x=\frac{4\sqrt{3}\left(3+\sqrt{2}\right)}{7}$$

Use the distributive property to multiply 4\sqrt{3} by 3+\sqrt{2}.

$$x=\frac{12\sqrt{3}+4\sqrt{3}\sqrt{2}}{7}$$

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

$$x=\frac{12\sqrt{3}+4\sqrt{6}}{7}$$

Divide each term of 12\sqrt{3}+4\sqrt{6} by 7 to get \frac{12}{7}\sqrt{3}+\frac{4}{7}\sqrt{6}.

$$x=\frac{12}{7}\sqrt{3}+\frac{4}{7}\sqrt{6}$$

Assign x

$$x≔\frac{4\sqrt{3}\left(\sqrt{2}+3\right)}{7}$$