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

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

Evaluate

$$13-4\sqrt{6}\approx 3.202041029$$

Solution Steps

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

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

Consider \left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{3}-\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Subtract 2 from 3 to get 1.

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

Anything divided by one gives itself.

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

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

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

The square of \sqrt{3} is 3.

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

Multiply 5 and 3 to get 15.

$$15-5\sqrt{3}\sqrt{2}+\sqrt{2}\sqrt{3}-\left(\sqrt{2}\right)^{2}$$

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

$$15-5\sqrt{6}+\sqrt{2}\sqrt{3}-\left(\sqrt{2}\right)^{2}$$

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

$$15-5\sqrt{6}+\sqrt{6}-\left(\sqrt{2}\right)^{2}$$

Combine -5\sqrt{6} and \sqrt{6} to get -4\sqrt{6}.

$$15-4\sqrt{6}-\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$15-4\sqrt{6}-2$$

Subtract 2 from 15 to get 13.

$$13-4\sqrt{6}$$