Solve 15/sqrt{6+1}+4/sqrt{6-2}-12/3-sqrt{6}(11+sqrt{6})

$$\frac{ 15 }{ \sqrt{ 6 } +1 } + \frac{ 4 }{ \sqrt{ 6 } -2 } - \frac{ 12 }{ 3- \sqrt{ 6 } } (11+ \sqrt{ 6 } )$$

Evaluate

$$-51\sqrt{6}-155\approx -279.923976882$$

Solution Steps

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

$$\frac{15\left(\sqrt{6}-1\right)}{\left(\sqrt{6}+1\right)\left(\sqrt{6}-1\right)}+\frac{4}{\sqrt{6}-2}-\frac{12}{3-\sqrt{6}}\left(11+\sqrt{6}\right)$$

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}.

$$\frac{15\left(\sqrt{6}-1\right)}{\left(\sqrt{6}\right)^{2}-1^{2}}+\frac{4}{\sqrt{6}-2}-\frac{12}{3-\sqrt{6}}\left(11+\sqrt{6}\right)$$

Square \sqrt{6}. Square 1.

$$\frac{15\left(\sqrt{6}-1\right)}{6-1}+\frac{4}{\sqrt{6}-2}-\frac{12}{3-\sqrt{6}}\left(11+\sqrt{6}\right)$$

Subtract 1 from 6 to get 5.

$$\frac{15\left(\sqrt{6}-1\right)}{5}+\frac{4}{\sqrt{6}-2}-\frac{12}{3-\sqrt{6}}\left(11+\sqrt{6}\right)$$

Divide 15\left(\sqrt{6}-1\right) by 5 to get 3\left(\sqrt{6}-1\right).

$$3\left(\sqrt{6}-1\right)+\frac{4}{\sqrt{6}-2}-\frac{12}{3-\sqrt{6}}\left(11+\sqrt{6}\right)$$

Rationalize the denominator of \frac{4}{\sqrt{6}-2} by multiplying numerator and denominator by \sqrt{6}+2.

$$3\left(\sqrt{6}-1\right)+\frac{4\left(\sqrt{6}+2\right)}{\left(\sqrt{6}-2\right)\left(\sqrt{6}+2\right)}-\frac{12}{3-\sqrt{6}}\left(11+\sqrt{6}\right)$$

Consider \left(\sqrt{6}-2\right)\left(\sqrt{6}+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}.

$$3\left(\sqrt{6}-1\right)+\frac{4\left(\sqrt{6}+2\right)}{\left(\sqrt{6}\right)^{2}-2^{2}}-\frac{12}{3-\sqrt{6}}\left(11+\sqrt{6}\right)$$

Square \sqrt{6}. Square 2.

$$3\left(\sqrt{6}-1\right)+\frac{4\left(\sqrt{6}+2\right)}{6-4}-\frac{12}{3-\sqrt{6}}\left(11+\sqrt{6}\right)$$

Subtract 4 from 6 to get 2.

$$3\left(\sqrt{6}-1\right)+\frac{4\left(\sqrt{6}+2\right)}{2}-\frac{12}{3-\sqrt{6}}\left(11+\sqrt{6}\right)$$

Divide 4\left(\sqrt{6}+2\right) by 2 to get 2\left(\sqrt{6}+2\right).

$$3\left(\sqrt{6}-1\right)+2\left(\sqrt{6}+2\right)-\frac{12}{3-\sqrt{6}}\left(11+\sqrt{6}\right)$$

Rationalize the denominator of \frac{12}{3-\sqrt{6}} by multiplying numerator and denominator by 3+\sqrt{6}.

$$3\left(\sqrt{6}-1\right)+2\left(\sqrt{6}+2\right)-\frac{12\left(3+\sqrt{6}\right)}{\left(3-\sqrt{6}\right)\left(3+\sqrt{6}\right)}\left(11+\sqrt{6}\right)$$

Consider \left(3-\sqrt{6}\right)\left(3+\sqrt{6}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$3\left(\sqrt{6}-1\right)+2\left(\sqrt{6}+2\right)-\frac{12\left(3+\sqrt{6}\right)}{3^{2}-\left(\sqrt{6}\right)^{2}}\left(11+\sqrt{6}\right)$$

Square 3. Square \sqrt{6}.

$$3\left(\sqrt{6}-1\right)+2\left(\sqrt{6}+2\right)-\frac{12\left(3+\sqrt{6}\right)}{9-6}\left(11+\sqrt{6}\right)$$

Subtract 6 from 9 to get 3.

$$3\left(\sqrt{6}-1\right)+2\left(\sqrt{6}+2\right)-\frac{12\left(3+\sqrt{6}\right)}{3}\left(11+\sqrt{6}\right)$$

Divide 12\left(3+\sqrt{6}\right) by 3 to get 4\left(3+\sqrt{6}\right).

$$3\left(\sqrt{6}-1\right)+2\left(\sqrt{6}+2\right)-4\left(3+\sqrt{6}\right)\left(11+\sqrt{6}\right)$$

Use the distributive property to multiply 3 by \sqrt{6}-1.

$$3\sqrt{6}-3+2\left(\sqrt{6}+2\right)-4\left(3+\sqrt{6}\right)\left(11+\sqrt{6}\right)$$

Use the distributive property to multiply 2 by \sqrt{6}+2.

$$3\sqrt{6}-3+2\sqrt{6}+4-4\left(3+\sqrt{6}\right)\left(11+\sqrt{6}\right)$$

Combine 3\sqrt{6} and 2\sqrt{6} to get 5\sqrt{6}.

$$5\sqrt{6}-3+4-4\left(3+\sqrt{6}\right)\left(11+\sqrt{6}\right)$$

Add -3 and 4 to get 1.

$$5\sqrt{6}+1-4\left(3+\sqrt{6}\right)\left(11+\sqrt{6}\right)$$

Use the distributive property to multiply -4 by 3+\sqrt{6}.

$$5\sqrt{6}+1+\left(-12-4\sqrt{6}\right)\left(11+\sqrt{6}\right)$$

Apply the distributive property by multiplying each term of -12-4\sqrt{6} by each term of 11+\sqrt{6}.

$$5\sqrt{6}+1-132-12\sqrt{6}-44\sqrt{6}-4\left(\sqrt{6}\right)^{2}$$

Combine -12\sqrt{6} and -44\sqrt{6} to get -56\sqrt{6}.

$$5\sqrt{6}+1-132-56\sqrt{6}-4\left(\sqrt{6}\right)^{2}$$

The square of \sqrt{6} is 6.

$$5\sqrt{6}+1-132-56\sqrt{6}-4\times 6$$

Multiply -4 and 6 to get -24.

$$5\sqrt{6}+1-132-56\sqrt{6}-24$$

Subtract 24 from -132 to get -156.

$$5\sqrt{6}+1-156-56\sqrt{6}$$

Subtract 156 from 1 to get -155.

$$5\sqrt{6}-155-56\sqrt{6}$$

Combine 5\sqrt{6} and -56\sqrt{6} to get -51\sqrt{6}.

$$-51\sqrt{6}-155$$