Solve 2/4-sqrt{5}*frac{4+sqrt{5}}{4+sqrt{5}}=frac{2(4+sqrt{5)}}{16-5}=frac{8+2sqrt{5}}{11}

$$\frac{ 2 }{ 4- \sqrt{ 5 } } \times \frac{ 4+ \sqrt{ 5 } }{ 4+ \sqrt{ 5 } } = \frac{ 2(4+ \sqrt{ 5) } }{ 16-5 } = \frac{ 8+2 \sqrt{ 5 } }{ 11 }$$

Verify

Solution Steps

Divide 4+\sqrt{5} by 4+\sqrt{5} to get 1.

$$\frac{2}{4-\sqrt{5}}\times 1=\frac{2\left(4+\sqrt{5}\right)}{16-5}\text{ and }\frac{2\left(4+\sqrt{5}\right)}{16-5}=\frac{8+2\sqrt{5}}{11}$$

Rationalize the denominator of \frac{2}{4-\sqrt{5}} by multiplying numerator and denominator by 4+\sqrt{5}.

$$\frac{2\left(4+\sqrt{5}\right)}{\left(4-\sqrt{5}\right)\left(4+\sqrt{5}\right)}\times 1=\frac{2\left(4+\sqrt{5}\right)}{16-5}\text{ and }\frac{2\left(4+\sqrt{5}\right)}{16-5}=\frac{8+2\sqrt{5}}{11}$$

Consider \left(4-\sqrt{5}\right)\left(4+\sqrt{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}.

$$\frac{2\left(4+\sqrt{5}\right)}{4^{2}-\left(\sqrt{5}\right)^{2}}\times 1=\frac{2\left(4+\sqrt{5}\right)}{16-5}\text{ and }\frac{2\left(4+\sqrt{5}\right)}{16-5}=\frac{8+2\sqrt{5}}{11}$$

Square 4. Square \sqrt{5}.

$$\frac{2\left(4+\sqrt{5}\right)}{16-5}\times 1=\frac{2\left(4+\sqrt{5}\right)}{16-5}\text{ and }\frac{2\left(4+\sqrt{5}\right)}{16-5}=\frac{8+2\sqrt{5}}{11}$$

Subtract 5 from 16 to get 11.

$$\frac{2\left(4+\sqrt{5}\right)}{11}\times 1=\frac{2\left(4+\sqrt{5}\right)}{16-5}\text{ and }\frac{2\left(4+\sqrt{5}\right)}{16-5}=\frac{8+2\sqrt{5}}{11}$$

Express \frac{2\left(4+\sqrt{5}\right)}{11}\times 1 as a single fraction.

$$\frac{2\left(4+\sqrt{5}\right)}{11}=\frac{2\left(4+\sqrt{5}\right)}{16-5}\text{ and }\frac{2\left(4+\sqrt{5}\right)}{16-5}=\frac{8+2\sqrt{5}}{11}$$

Subtract 5 from 16 to get 11.

$$\frac{2\left(4+\sqrt{5}\right)}{11}=\frac{2\left(4+\sqrt{5}\right)}{11}\text{ and }\frac{2\left(4+\sqrt{5}\right)}{16-5}=\frac{8+2\sqrt{5}}{11}$$

Subtract 5 from 16 to get 11.

$$\frac{2\left(4+\sqrt{5}\right)}{11}=\frac{2\left(4+\sqrt{5}\right)}{11}\text{ and }\frac{2\left(4+\sqrt{5}\right)}{11}=\frac{8+2\sqrt{5}}{11}$$

Use the distributive property to multiply 2 by 4+\sqrt{5}.

$$\frac{8+2\sqrt{5}}{11}=\frac{2\left(4+\sqrt{5}\right)}{11}\text{ and }\frac{2\left(4+\sqrt{5}\right)}{11}=\frac{8+2\sqrt{5}}{11}$$

Use the distributive property to multiply 2 by 4+\sqrt{5}.

$$\frac{8+2\sqrt{5}}{11}=\frac{8+2\sqrt{5}}{11}\text{ and }\frac{2\left(4+\sqrt{5}\right)}{11}=\frac{8+2\sqrt{5}}{11}$$

Subtract \frac{8+2\sqrt{5}}{11} from both sides.

$$\frac{8+2\sqrt{5}}{11}-\frac{8+2\sqrt{5}}{11}=0\text{ and }\frac{2\left(4+\sqrt{5}\right)}{11}=\frac{8+2\sqrt{5}}{11}$$

Combine \frac{8+2\sqrt{5}}{11} and -\frac{8+2\sqrt{5}}{11} to get 0.

$$0=0\text{ and }\frac{2\left(4+\sqrt{5}\right)}{11}=\frac{8+2\sqrt{5}}{11}$$

Compare 0 and 0.

$$\text{true}\text{ and }\frac{2\left(4+\sqrt{5}\right)}{11}=\frac{8+2\sqrt{5}}{11}$$

Use the distributive property to multiply 2 by 4+\sqrt{5}.

$$\text{true}\text{ and }\frac{8+2\sqrt{5}}{11}=\frac{8+2\sqrt{5}}{11}$$

Subtract \frac{8+2\sqrt{5}}{11} from both sides.

$$\text{true}\text{ and }\frac{8+2\sqrt{5}}{11}-\frac{8+2\sqrt{5}}{11}=0$$

Combine \frac{8+2\sqrt{5}}{11} and -\frac{8+2\sqrt{5}}{11} to get 0.

$$\text{true}\text{ and }0=0$$

Compare 0 and 0.

$$\text{true}\text{ and }\text{true}$$

The conjunction of \text{true} and \text{true} is \text{true}.

$$\text{true}$$