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

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

Evaluate

$$5-\sqrt{6}\approx 2.550510257$$

Solution Steps

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

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

Consider \left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{2}-\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}.

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

Square \sqrt{2}. Square \sqrt{3}.

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

Subtract 3 from 2 to get -1.

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

Anything divided by -1 gives its opposite.

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

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

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

The square of \sqrt{2} is 2.

$$-\left(2\times 2-2\sqrt{3}\sqrt{2}+3\sqrt{3}\sqrt{2}-3\left(\sqrt{3}\right)^{2}\right)$$

Multiply 2 and 2 to get 4.

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

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

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

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

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

Combine -2\sqrt{6} and 3\sqrt{6} to get \sqrt{6}.

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

The square of \sqrt{3} is 3.

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

Multiply -3 and 3 to get -9.

$$-\left(4+\sqrt{6}-9\right)$$

Subtract 9 from 4 to get -5.

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

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

$$-\left(-5\right)-\sqrt{6}$$

The opposite of -5 is 5.

$$5-\sqrt{6}$$