Solve 9/sqrt{7-2}-4/3+sqrt{7}+5/sqrt{6-sqrt{7}}

$$\frac { 9 } { \sqrt { 7 } - 2 } - \frac { 4 } { 3 + \sqrt { 7 } } + \frac { 5 } { \sqrt { 6 } - \sqrt { 7 } }$$

Evaluate

$$-5\sqrt{6}\approx -12.247448714$$

Solution Steps

Rationalize the denominator of \frac{9}{\sqrt{7}-2} by multiplying numerator and denominator by \sqrt{7}+2.

$$\frac{9\left(\sqrt{7}+2\right)}{\left(\sqrt{7}-2\right)\left(\sqrt{7}+2\right)}-\frac{4}{3+\sqrt{7}}+\frac{5}{\sqrt{6}-\sqrt{7}}$$

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}.

$$\frac{9\left(\sqrt{7}+2\right)}{\left(\sqrt{7}\right)^{2}-2^{2}}-\frac{4}{3+\sqrt{7}}+\frac{5}{\sqrt{6}-\sqrt{7}}$$

Square \sqrt{7}. Square 2.

$$\frac{9\left(\sqrt{7}+2\right)}{7-4}-\frac{4}{3+\sqrt{7}}+\frac{5}{\sqrt{6}-\sqrt{7}}$$

Subtract 4 from 7 to get 3.

$$\frac{9\left(\sqrt{7}+2\right)}{3}-\frac{4}{3+\sqrt{7}}+\frac{5}{\sqrt{6}-\sqrt{7}}$$

Divide 9\left(\sqrt{7}+2\right) by 3 to get 3\left(\sqrt{7}+2\right).

$$3\left(\sqrt{7}+2\right)-\frac{4}{3+\sqrt{7}}+\frac{5}{\sqrt{6}-\sqrt{7}}$$

Rationalize the denominator of \frac{4}{3+\sqrt{7}} by multiplying numerator and denominator by 3-\sqrt{7}.

$$3\left(\sqrt{7}+2\right)-\frac{4\left(3-\sqrt{7}\right)}{\left(3+\sqrt{7}\right)\left(3-\sqrt{7}\right)}+\frac{5}{\sqrt{6}-\sqrt{7}}$$

Consider \left(3+\sqrt{7}\right)\left(3-\sqrt{7}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$3\left(\sqrt{7}+2\right)-\frac{4\left(3-\sqrt{7}\right)}{3^{2}-\left(\sqrt{7}\right)^{2}}+\frac{5}{\sqrt{6}-\sqrt{7}}$$

Square 3. Square \sqrt{7}.

$$3\left(\sqrt{7}+2\right)-\frac{4\left(3-\sqrt{7}\right)}{9-7}+\frac{5}{\sqrt{6}-\sqrt{7}}$$

Subtract 7 from 9 to get 2.

$$3\left(\sqrt{7}+2\right)-\frac{4\left(3-\sqrt{7}\right)}{2}+\frac{5}{\sqrt{6}-\sqrt{7}}$$

Divide 4\left(3-\sqrt{7}\right) by 2 to get 2\left(3-\sqrt{7}\right).

$$3\left(\sqrt{7}+2\right)-2\left(3-\sqrt{7}\right)+\frac{5}{\sqrt{6}-\sqrt{7}}$$

Rationalize the denominator of \frac{5}{\sqrt{6}-\sqrt{7}} by multiplying numerator and denominator by \sqrt{6}+\sqrt{7}.

$$3\left(\sqrt{7}+2\right)-2\left(3-\sqrt{7}\right)+\frac{5\left(\sqrt{6}+\sqrt{7}\right)}{\left(\sqrt{6}-\sqrt{7}\right)\left(\sqrt{6}+\sqrt{7}\right)}$$

Consider \left(\sqrt{6}-\sqrt{7}\right)\left(\sqrt{6}+\sqrt{7}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$3\left(\sqrt{7}+2\right)-2\left(3-\sqrt{7}\right)+\frac{5\left(\sqrt{6}+\sqrt{7}\right)}{\left(\sqrt{6}\right)^{2}-\left(\sqrt{7}\right)^{2}}$$

Square \sqrt{6}. Square \sqrt{7}.

$$3\left(\sqrt{7}+2\right)-2\left(3-\sqrt{7}\right)+\frac{5\left(\sqrt{6}+\sqrt{7}\right)}{6-7}$$

Subtract 7 from 6 to get -1.

$$3\left(\sqrt{7}+2\right)-2\left(3-\sqrt{7}\right)+\frac{5\left(\sqrt{6}+\sqrt{7}\right)}{-1}$$

Anything divided by -1 gives its opposite.

$$3\left(\sqrt{7}+2\right)-2\left(3-\sqrt{7}\right)-5\left(\sqrt{6}+\sqrt{7}\right)$$

Use the distributive property to multiply 3 by \sqrt{7}+2.

$$3\sqrt{7}+6-2\left(3-\sqrt{7}\right)-5\left(\sqrt{6}+\sqrt{7}\right)$$

Use the distributive property to multiply 2 by 3-\sqrt{7}.

$$3\sqrt{7}+6-\left(6-2\sqrt{7}\right)-5\left(\sqrt{6}+\sqrt{7}\right)$$

To find the opposite of 6-2\sqrt{7}, find the opposite of each term.

$$3\sqrt{7}+6-6-\left(-2\sqrt{7}\right)-5\left(\sqrt{6}+\sqrt{7}\right)$$

The opposite of -2\sqrt{7} is 2\sqrt{7}.

$$3\sqrt{7}+6-6+2\sqrt{7}-5\left(\sqrt{6}+\sqrt{7}\right)$$

Subtract 6 from 6 to get 0.

$$3\sqrt{7}+2\sqrt{7}-5\left(\sqrt{6}+\sqrt{7}\right)$$

Combine 3\sqrt{7} and 2\sqrt{7} to get 5\sqrt{7}.

$$5\sqrt{7}-5\left(\sqrt{6}+\sqrt{7}\right)$$

Use the distributive property to multiply -5 by \sqrt{6}+\sqrt{7}.

$$5\sqrt{7}-5\sqrt{6}-5\sqrt{7}$$

Combine 5\sqrt{7} and -5\sqrt{7} to get 0.

$$-5\sqrt{6}$$