Solve frac{2sqrt{2}-3}{2+sqrt{5}}

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

Evaluate

$$2\sqrt{10}+6-3\sqrt{5}-4\sqrt{2}\approx -0.040502862$$

Solution Steps

Rationalize the denominator of \frac{2\sqrt{2}-3}{2+\sqrt{5}} by multiplying numerator and denominator by 2-\sqrt{5}.

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

Consider \left(2+\sqrt{5}\right)\left(2-\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}.

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

Square 2. Square \sqrt{5}.

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

Subtract 5 from 4 to get -1.

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

Anything divided by -1 gives its opposite.

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

Apply the distributive property by multiplying each term of 2\sqrt{2}-3 by each term of 2-\sqrt{5}.

$$-\left(4\sqrt{2}-2\sqrt{2}\sqrt{5}-6+3\sqrt{5}\right)$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$-\left(4\sqrt{2}-2\sqrt{10}-6+3\sqrt{5}\right)$$

To find the opposite of 4\sqrt{2}-2\sqrt{10}-6+3\sqrt{5}, find the opposite of each term.

$$-4\sqrt{2}-\left(-2\sqrt{10}\right)-\left(-6\right)-3\sqrt{5}$$

The opposite of -2\sqrt{10} is 2\sqrt{10}.

$$-4\sqrt{2}+2\sqrt{10}-\left(-6\right)-3\sqrt{5}$$

The opposite of -6 is 6.

$$-4\sqrt{2}+2\sqrt{10}+6-3\sqrt{5}$$