Solve frac{sqrt{10}+sqrt{2}+sqrt{3}}{sqrt{7}-1}

$$\frac { \sqrt { 10 } + \sqrt { 2 } + \sqrt { 3 } } { \sqrt { 7 } - 1 }$$

Evaluate

$$\frac{\sqrt{2}+\sqrt{3}+\sqrt{10}+\sqrt{14}+\sqrt{21}+\sqrt{70}}{6}\approx 3.83322923$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{10}+\sqrt{2}+\sqrt{3}}{\sqrt{7}-1} by multiplying numerator and denominator by \sqrt{7}+1.

$$\frac{\left(\sqrt{10}+\sqrt{2}+\sqrt{3}\right)\left(\sqrt{7}+1\right)}{\left(\sqrt{7}-1\right)\left(\sqrt{7}+1\right)}$$

Consider \left(\sqrt{7}-1\right)\left(\sqrt{7}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{10}+\sqrt{2}+\sqrt{3}\right)\left(\sqrt{7}+1\right)}{\left(\sqrt{7}\right)^{2}-1^{2}}$$

Square \sqrt{7}. Square 1.

$$\frac{\left(\sqrt{10}+\sqrt{2}+\sqrt{3}\right)\left(\sqrt{7}+1\right)}{7-1}$$

Subtract 1 from 7 to get 6.

$$\frac{\left(\sqrt{10}+\sqrt{2}+\sqrt{3}\right)\left(\sqrt{7}+1\right)}{6}$$

Apply the distributive property by multiplying each term of \sqrt{10}+\sqrt{2}+\sqrt{3} by each term of \sqrt{7}+1.

$$\frac{\sqrt{10}\sqrt{7}+\sqrt{10}+\sqrt{2}\sqrt{7}+\sqrt{2}+\sqrt{3}\sqrt{7}+\sqrt{3}}{6}$$

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

$$\frac{\sqrt{70}+\sqrt{10}+\sqrt{2}\sqrt{7}+\sqrt{2}+\sqrt{3}\sqrt{7}+\sqrt{3}}{6}$$

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

$$\frac{\sqrt{70}+\sqrt{10}+\sqrt{14}+\sqrt{2}+\sqrt{3}\sqrt{7}+\sqrt{3}}{6}$$

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

$$\frac{\sqrt{70}+\sqrt{10}+\sqrt{14}+\sqrt{2}+\sqrt{21}+\sqrt{3}}{6}$$