Solve frac{sqrt{3}}{3sqrt{3}-5}

$$\frac { \sqrt { 3 } } { 3 \sqrt { 3 } - 5 }$$

Evaluate

$$\frac{5\sqrt{3}+9}{2}\approx 8.830127019$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{3}}{3\sqrt{3}-5} by multiplying numerator and denominator by 3\sqrt{3}+5.

$$\frac{\sqrt{3}\left(3\sqrt{3}+5\right)}{\left(3\sqrt{3}-5\right)\left(3\sqrt{3}+5\right)}$$

Consider \left(3\sqrt{3}-5\right)\left(3\sqrt{3}+5\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\sqrt{3}\left(3\sqrt{3}+5\right)}{\left(3\sqrt{3}\right)^{2}-5^{2}}$$

Expand \left(3\sqrt{3}\right)^{2}.

$$\frac{\sqrt{3}\left(3\sqrt{3}+5\right)}{3^{2}\left(\sqrt{3}\right)^{2}-5^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\sqrt{3}\left(3\sqrt{3}+5\right)}{9\left(\sqrt{3}\right)^{2}-5^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{3}\left(3\sqrt{3}+5\right)}{9\times 3-5^{2}}$$

Multiply 9 and 3 to get 27.

$$\frac{\sqrt{3}\left(3\sqrt{3}+5\right)}{27-5^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{\sqrt{3}\left(3\sqrt{3}+5\right)}{27-25}$$

Subtract 25 from 27 to get 2.

$$\frac{\sqrt{3}\left(3\sqrt{3}+5\right)}{2}$$

Use the distributive property to multiply \sqrt{3} by 3\sqrt{3}+5.

$$\frac{3\left(\sqrt{3}\right)^{2}+5\sqrt{3}}{2}$$

The square of \sqrt{3} is 3.

$$\frac{3\times 3+5\sqrt{3}}{2}$$

Multiply 3 and 3 to get 9.

$$\frac{9+5\sqrt{3}}{2}$$