Solve 4/5+sqrt{3}=

$$\frac{ 4 }{ 5+ \sqrt{ 3 } } =$$

Evaluate

$$\frac{10-2\sqrt{3}}{11}\approx 0.59417258$$

Solution Steps

Rationalize the denominator of \frac{4}{5+\sqrt{3}} by multiplying numerator and denominator by 5-\sqrt{3}.

$$\frac{4\left(5-\sqrt{3}\right)}{\left(5+\sqrt{3}\right)\left(5-\sqrt{3}\right)}$$

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

$$\frac{4\left(5-\sqrt{3}\right)}{5^{2}-\left(\sqrt{3}\right)^{2}}$$

Square 5. Square \sqrt{3}.

$$\frac{4\left(5-\sqrt{3}\right)}{25-3}$$

Subtract 3 from 25 to get 22.

$$\frac{4\left(5-\sqrt{3}\right)}{22}$$

Divide 4\left(5-\sqrt{3}\right) by 22 to get \frac{2}{11}\left(5-\sqrt{3}\right).

$$\frac{2}{11}\left(5-\sqrt{3}\right)$$

Use the distributive property to multiply \frac{2}{11} by 5-\sqrt{3}.

$$\frac{2}{11}\times 5+\frac{2}{11}\left(-1\right)\sqrt{3}$$

Express \frac{2}{11}\times 5 as a single fraction.

$$\frac{2\times 5}{11}+\frac{2}{11}\left(-1\right)\sqrt{3}$$

Multiply 2 and 5 to get 10.

$$\frac{10}{11}+\frac{2}{11}\left(-1\right)\sqrt{3}$$

Multiply \frac{2}{11} and -1 to get -\frac{2}{11}.

$$\frac{10}{11}-\frac{2}{11}\sqrt{3}$$