Solve frac{sqrt{60}+sqrt{10}-sqrt{6}-1}{sqrt{6}+1}-sqrt{10}

$$\frac{ \sqrt{ 60 } + \sqrt{ 10 } - \sqrt{ 6 } -1 }{ \sqrt{ 6 } +1 } - \sqrt{ 10 }$$

Evaluate

$$-1$$

Short Solution Steps

Factor 60=2^{2}\times 15. Rewrite the square root of the product \sqrt{2^{2}\times 15} as the product of square roots \sqrt{2^{2}}\sqrt{15}. Take the square root of 2^{2}.

$$\frac{2\sqrt{15}+\sqrt{10}-\sqrt{6}-1}{\sqrt{6}+1}-\sqrt{10}$$

Rationalize the denominator of \frac{2\sqrt{15}+\sqrt{10}-\sqrt{6}-1}{\sqrt{6}+1} by multiplying numerator and denominator by \sqrt{6}-1.

$$\frac{\left(2\sqrt{15}+\sqrt{10}-\sqrt{6}-1\right)\left(\sqrt{6}-1\right)}{\left(\sqrt{6}+1\right)\left(\sqrt{6}-1\right)}-\sqrt{10}$$

Consider \left(\sqrt{6}+1\right)\left(\sqrt{6}-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}.

$$\frac{\left(2\sqrt{15}+\sqrt{10}-\sqrt{6}-1\right)\left(\sqrt{6}-1\right)}{\left(\sqrt{6}\right)^{2}-1^{2}}-\sqrt{10}$$

Square \sqrt{6}. Square 1.

$$\frac{\left(2\sqrt{15}+\sqrt{10}-\sqrt{6}-1\right)\left(\sqrt{6}-1\right)}{6-1}-\sqrt{10}$$

Subtract 1 from 6 to get 5.

$$\frac{\left(2\sqrt{15}+\sqrt{10}-\sqrt{6}-1\right)\left(\sqrt{6}-1\right)}{5}-\sqrt{10}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{10} times \frac{5}{5}.

$$\frac{\left(2\sqrt{15}+\sqrt{10}-\sqrt{6}-1\right)\left(\sqrt{6}-1\right)}{5}-\frac{5\sqrt{10}}{5}$$

Since \frac{\left(2\sqrt{15}+\sqrt{10}-\sqrt{6}-1\right)\left(\sqrt{6}-1\right)}{5} and \frac{5\sqrt{10}}{5} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(2\sqrt{15}+\sqrt{10}-\sqrt{6}-1\right)\left(\sqrt{6}-1\right)-5\sqrt{10}}{5}$$

Do the multiplications in \left(2\sqrt{15}+\sqrt{10}-\sqrt{6}-1\right)\left(\sqrt{6}-1\right)-5\sqrt{10}.

$$\frac{6\sqrt{10}-2\sqrt{15}+2\sqrt{15}-\sqrt{10}-6+\sqrt{6}-\sqrt{6}+1-5\sqrt{10}}{5}$$

Do the calculations in 6\sqrt{10}-2\sqrt{15}+2\sqrt{15}-\sqrt{10}-6+\sqrt{6}-\sqrt{6}+1-5\sqrt{10}.

$$\frac{-5}{5}$$

Divide -5 by 5 to get -1.

$$-1$$

Factor

$$-1$$