Solve 6/3-sqrt{7}+1/8+3sqrt{7}

$$\frac { 6 } { 3 - \sqrt { 7 } } + \frac { 1 } { 8 + 3 \sqrt { 7 } }$$

Evaluate

$$17$$

Solution Steps

Rationalize the denominator of \frac{6}{3-\sqrt{7}} by multiplying numerator and denominator by 3+\sqrt{7}.

$$\frac{6\left(3+\sqrt{7}\right)}{\left(3-\sqrt{7}\right)\left(3+\sqrt{7}\right)}+\frac{1}{8+3\sqrt{7}}$$

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}.

$$\frac{6\left(3+\sqrt{7}\right)}{3^{2}-\left(\sqrt{7}\right)^{2}}+\frac{1}{8+3\sqrt{7}}$$

Square 3. Square \sqrt{7}.

$$\frac{6\left(3+\sqrt{7}\right)}{9-7}+\frac{1}{8+3\sqrt{7}}$$

Subtract 7 from 9 to get 2.

$$\frac{6\left(3+\sqrt{7}\right)}{2}+\frac{1}{8+3\sqrt{7}}$$

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

$$3\left(3+\sqrt{7}\right)+\frac{1}{8+3\sqrt{7}}$$

Rationalize the denominator of \frac{1}{8+3\sqrt{7}} by multiplying numerator and denominator by 8-3\sqrt{7}.

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

Consider \left(8+3\sqrt{7}\right)\left(8-3\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}.

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

Calculate 8 to the power of 2 and get 64.

$$3\left(3+\sqrt{7}\right)+\frac{8-3\sqrt{7}}{64-\left(3\sqrt{7}\right)^{2}}$$

Expand \left(3\sqrt{7}\right)^{2}.

$$3\left(3+\sqrt{7}\right)+\frac{8-3\sqrt{7}}{64-3^{2}\left(\sqrt{7}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$3\left(3+\sqrt{7}\right)+\frac{8-3\sqrt{7}}{64-9\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$3\left(3+\sqrt{7}\right)+\frac{8-3\sqrt{7}}{64-9\times 7}$$

Multiply 9 and 7 to get 63.

$$3\left(3+\sqrt{7}\right)+\frac{8-3\sqrt{7}}{64-63}$$

Subtract 63 from 64 to get 1.

$$3\left(3+\sqrt{7}\right)+\frac{8-3\sqrt{7}}{1}$$

Anything divided by one gives itself.

$$3\left(3+\sqrt{7}\right)+8-3\sqrt{7}$$

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

$$9+3\sqrt{7}+8-3\sqrt{7}$$

Add 9 and 8 to get 17.

$$17+3\sqrt{7}-3\sqrt{7}$$

Combine 3\sqrt{7} and -3\sqrt{7} to get 0.

$$17$$

Factor

$$17$$