Solve 8/4-sqrt{6}

$$\frac{ 8 }{ 4- \sqrt{ 6 } }$$

Evaluate

$$\frac{4\left(\sqrt{6}+4\right)}{5}\approx 5.159591794$$

Solution Steps

Rationalize the denominator of \frac{8}{4-\sqrt{6}} by multiplying numerator and denominator by 4+\sqrt{6}.

$$\frac{8\left(4+\sqrt{6}\right)}{\left(4-\sqrt{6}\right)\left(4+\sqrt{6}\right)}$$

Consider \left(4-\sqrt{6}\right)\left(4+\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}.

$$\frac{8\left(4+\sqrt{6}\right)}{4^{2}-\left(\sqrt{6}\right)^{2}}$$

Square 4. Square \sqrt{6}.

$$\frac{8\left(4+\sqrt{6}\right)}{16-6}$$

Subtract 6 from 16 to get 10.

$$\frac{8\left(4+\sqrt{6}\right)}{10}$$

Divide 8\left(4+\sqrt{6}\right) by 10 to get \frac{4}{5}\left(4+\sqrt{6}\right).

$$\frac{4}{5}\left(4+\sqrt{6}\right)$$

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

$$\frac{4}{5}\times 4+\frac{4}{5}\sqrt{6}$$

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

$$\frac{4\times 4}{5}+\frac{4}{5}\sqrt{6}$$

Multiply 4 and 4 to get 16.

$$\frac{16}{5}+\frac{4}{5}\sqrt{6}$$