Solve frac{sqrt{10}+sqrt{2}+sqrt{3}}{sqrt{7}-2}

$$\frac { \sqrt { 10 } + \sqrt { 2 } + \sqrt { 3 } } { \sqrt { 7 } - 2 }$$

Evaluate

$$\frac{\sqrt{14}+\sqrt{21}+\sqrt{70}+2\sqrt{2}+2\sqrt{3}+2\sqrt{10}}{3}\approx 9.769305802$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{10}+\sqrt{2}+\sqrt{3}}{\sqrt{7}-2} by multiplying numerator and denominator by \sqrt{7}+2.

$$\frac{\left(\sqrt{10}+\sqrt{2}+\sqrt{3}\right)\left(\sqrt{7}+2\right)}{\left(\sqrt{7}-2\right)\left(\sqrt{7}+2\right)}$$

Consider \left(\sqrt{7}-2\right)\left(\sqrt{7}+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}.

$$\frac{\left(\sqrt{10}+\sqrt{2}+\sqrt{3}\right)\left(\sqrt{7}+2\right)}{\left(\sqrt{7}\right)^{2}-2^{2}}$$

Square \sqrt{7}. Square 2.

$$\frac{\left(\sqrt{10}+\sqrt{2}+\sqrt{3}\right)\left(\sqrt{7}+2\right)}{7-4}$$

Subtract 4 from 7 to get 3.

$$\frac{\left(\sqrt{10}+\sqrt{2}+\sqrt{3}\right)\left(\sqrt{7}+2\right)}{3}$$

Apply the distributive property by multiplying each term of \sqrt{10}+\sqrt{2}+\sqrt{3} by each term of \sqrt{7}+2.

$$\frac{\sqrt{10}\sqrt{7}+2\sqrt{10}+\sqrt{2}\sqrt{7}+2\sqrt{2}+\sqrt{3}\sqrt{7}+2\sqrt{3}}{3}$$

To multiply \sqrt{10} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{\sqrt{70}+2\sqrt{10}+\sqrt{2}\sqrt{7}+2\sqrt{2}+\sqrt{3}\sqrt{7}+2\sqrt{3}}{3}$$

To multiply \sqrt{2} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{\sqrt{70}+2\sqrt{10}+\sqrt{14}+2\sqrt{2}+\sqrt{3}\sqrt{7}+2\sqrt{3}}{3}$$

To multiply \sqrt{3} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{\sqrt{70}+2\sqrt{10}+\sqrt{14}+2\sqrt{2}+\sqrt{21}+2\sqrt{3}}{3}$$