Solve 8/3-sqrt{7}-6/2sqrt{7-5}

$$\frac { 8 } { 3 - \sqrt { 7 } } - \frac { 6 } { 2 \sqrt { 7 } - 5 }$$

Evaluate

$$2$$

Solution Steps

Rationalize the denominator of \frac{8}{3-\sqrt{7}} by multiplying numerator and denominator by 3+\sqrt{7}.

$$\frac{8\left(3+\sqrt{7}\right)}{\left(3-\sqrt{7}\right)\left(3+\sqrt{7}\right)}-\frac{6}{2\sqrt{7}-5}$$

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}.

$$\frac{8\left(3+\sqrt{7}\right)}{3^{2}-\left(\sqrt{7}\right)^{2}}-\frac{6}{2\sqrt{7}-5}$$

Square 3. Square \sqrt{7}.

$$\frac{8\left(3+\sqrt{7}\right)}{9-7}-\frac{6}{2\sqrt{7}-5}$$

Subtract 7 from 9 to get 2.

$$\frac{8\left(3+\sqrt{7}\right)}{2}-\frac{6}{2\sqrt{7}-5}$$

Divide 8\left(3+\sqrt{7}\right) by 2 to get 4\left(3+\sqrt{7}\right).

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

Rationalize the denominator of \frac{6}{2\sqrt{7}-5} by multiplying numerator and denominator by 2\sqrt{7}+5.

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

Consider \left(2\sqrt{7}-5\right)\left(2\sqrt{7}+5\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(2\sqrt{7}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{7} is 7.

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

Multiply 4 and 7 to get 28.

$$4\left(3+\sqrt{7}\right)-\frac{6\left(2\sqrt{7}+5\right)}{28-5^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$4\left(3+\sqrt{7}\right)-\frac{6\left(2\sqrt{7}+5\right)}{28-25}$$

Subtract 25 from 28 to get 3.

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

Divide 6\left(2\sqrt{7}+5\right) by 3 to get 2\left(2\sqrt{7}+5\right).

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

Use the distributive property to multiply 4 by 3+\sqrt{7}.

$$12+4\sqrt{7}-2\left(2\sqrt{7}+5\right)$$

Use the distributive property to multiply 2 by 2\sqrt{7}+5.

$$12+4\sqrt{7}-\left(4\sqrt{7}+10\right)$$

To find the opposite of 4\sqrt{7}+10, find the opposite of each term.

$$12+4\sqrt{7}-4\sqrt{7}-10$$

Combine 4\sqrt{7} and -4\sqrt{7} to get 0.

$$12-10$$

Subtract 10 from 12 to get 2.

$$2$$

Factor

$$2$$