Solve frac{sqrt{7}+1}{sqrt{7}-1}*frac{sqrt{7}+1}{sqrt{7}-1}

$$\frac { \sqrt { 7 } + 1 } { \sqrt { 7 } - 1 } * \frac { \sqrt { 7 } + 1 } { \sqrt { 7 } - 1 }$$

Evaluate

$$\frac{8\sqrt{7}+23}{9}\approx 4.907334499$$

Solution Steps

Multiply \frac{\sqrt{7}+1}{\sqrt{7}-1} and \frac{\sqrt{7}+1}{\sqrt{7}-1} to get \left(\frac{\sqrt{7}+1}{\sqrt{7}-1}\right)^{2}.

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

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

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

Consider \left(\sqrt{7}-1\right)\left(\sqrt{7}+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}.

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

Square \sqrt{7}. Square 1.

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

Subtract 1 from 7 to get 6.

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

Multiply \sqrt{7}+1 and \sqrt{7}+1 to get \left(\sqrt{7}+1\right)^{2}.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{7}+1\right)^{2}.

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

The square of \sqrt{7} is 7.

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

Add 7 and 1 to get 8.

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

To raise \frac{8+2\sqrt{7}}{6} to a power, raise both numerator and denominator to the power and then divide.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(8+2\sqrt{7}\right)^{2}.

$$\frac{64+32\sqrt{7}+4\left(\sqrt{7}\right)^{2}}{6^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{64+32\sqrt{7}+4\times 7}{6^{2}}$$

Multiply 4 and 7 to get 28.

$$\frac{64+32\sqrt{7}+28}{6^{2}}$$

Add 64 and 28 to get 92.

$$\frac{92+32\sqrt{7}}{6^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{92+32\sqrt{7}}{36}$$

Factor

$$\frac{8 \sqrt{7} + 23}{9} = 4.90733449872408$$