Solve (21-8sqrt{5})div(4-sqrt{5})

$$(21-8 \sqrt{ 5 } ) \div (4- \sqrt{ 5 } )$$

Evaluate

$$4-\sqrt{5}\approx 1.763932023$$

Solution Steps

Rationalize the denominator of \frac{21-8\sqrt{5}}{4-\sqrt{5}} by multiplying numerator and denominator by 4+\sqrt{5}.

$$\frac{\left(21-8\sqrt{5}\right)\left(4+\sqrt{5}\right)}{\left(4-\sqrt{5}\right)\left(4+\sqrt{5}\right)}$$

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}.

$$\frac{\left(21-8\sqrt{5}\right)\left(4+\sqrt{5}\right)}{4^{2}-\left(\sqrt{5}\right)^{2}}$$

Square 4. Square \sqrt{5}.

$$\frac{\left(21-8\sqrt{5}\right)\left(4+\sqrt{5}\right)}{16-5}$$

Subtract 5 from 16 to get 11.

$$\frac{\left(21-8\sqrt{5}\right)\left(4+\sqrt{5}\right)}{11}$$

Apply the distributive property by multiplying each term of 21-8\sqrt{5} by each term of 4+\sqrt{5}.

$$\frac{84+21\sqrt{5}-32\sqrt{5}-8\left(\sqrt{5}\right)^{2}}{11}$$

Combine 21\sqrt{5} and -32\sqrt{5} to get -11\sqrt{5}.

$$\frac{84-11\sqrt{5}-8\left(\sqrt{5}\right)^{2}}{11}$$

The square of \sqrt{5} is 5.

$$\frac{84-11\sqrt{5}-8\times 5}{11}$$

Multiply -8 and 5 to get -40.

$$\frac{84-11\sqrt{5}-40}{11}$$

Subtract 40 from 84 to get 44.

$$\frac{44-11\sqrt{5}}{11}$$

Divide each term of 44-11\sqrt{5} by 11 to get 4-\sqrt{5}.

$$4-\sqrt{5}$$