Solve frac{8sqrt{11}+7sqrt{2}}{8sqrt{11}-7sqrt{2}}

$$\frac { 8 \sqrt { 11 } + 7 \sqrt { 2 } } { 8 \sqrt { 11 } - 7 \sqrt { 2 } }$$

Evaluate

$$\frac{56\sqrt{22}+401}{303}\approx 2.190307863$$

Solution Steps

Rationalize the denominator of \frac{8\sqrt{11}+7\sqrt{2}}{8\sqrt{11}-7\sqrt{2}} by multiplying numerator and denominator by 8\sqrt{11}+7\sqrt{2}.

$$\frac{\left(8\sqrt{11}+7\sqrt{2}\right)\left(8\sqrt{11}+7\sqrt{2}\right)}{\left(8\sqrt{11}-7\sqrt{2}\right)\left(8\sqrt{11}+7\sqrt{2}\right)}$$

Consider \left(8\sqrt{11}-7\sqrt{2}\right)\left(8\sqrt{11}+7\sqrt{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}.

$$\frac{\left(8\sqrt{11}+7\sqrt{2}\right)\left(8\sqrt{11}+7\sqrt{2}\right)}{\left(8\sqrt{11}\right)^{2}-\left(-7\sqrt{2}\right)^{2}}$$

Multiply 8\sqrt{11}+7\sqrt{2} and 8\sqrt{11}+7\sqrt{2} to get \left(8\sqrt{11}+7\sqrt{2}\right)^{2}.

$$\frac{\left(8\sqrt{11}+7\sqrt{2}\right)^{2}}{\left(8\sqrt{11}\right)^{2}-\left(-7\sqrt{2}\right)^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(8\sqrt{11}+7\sqrt{2}\right)^{2}.

$$\frac{64\left(\sqrt{11}\right)^{2}+112\sqrt{11}\sqrt{2}+49\left(\sqrt{2}\right)^{2}}{\left(8\sqrt{11}\right)^{2}-\left(-7\sqrt{2}\right)^{2}}$$

The square of \sqrt{11} is 11.

$$\frac{64\times 11+112\sqrt{11}\sqrt{2}+49\left(\sqrt{2}\right)^{2}}{\left(8\sqrt{11}\right)^{2}-\left(-7\sqrt{2}\right)^{2}}$$

Multiply 64 and 11 to get 704.

$$\frac{704+112\sqrt{11}\sqrt{2}+49\left(\sqrt{2}\right)^{2}}{\left(8\sqrt{11}\right)^{2}-\left(-7\sqrt{2}\right)^{2}}$$

To multiply \sqrt{11} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{704+112\sqrt{22}+49\left(\sqrt{2}\right)^{2}}{\left(8\sqrt{11}\right)^{2}-\left(-7\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{704+112\sqrt{22}+49\times 2}{\left(8\sqrt{11}\right)^{2}-\left(-7\sqrt{2}\right)^{2}}$$

Multiply 49 and 2 to get 98.

$$\frac{704+112\sqrt{22}+98}{\left(8\sqrt{11}\right)^{2}-\left(-7\sqrt{2}\right)^{2}}$$

Add 704 and 98 to get 802.

$$\frac{802+112\sqrt{22}}{\left(8\sqrt{11}\right)^{2}-\left(-7\sqrt{2}\right)^{2}}$$

Expand \left(8\sqrt{11}\right)^{2}.

$$\frac{802+112\sqrt{22}}{8^{2}\left(\sqrt{11}\right)^{2}-\left(-7\sqrt{2}\right)^{2}}$$

Calculate 8 to the power of 2 and get 64.

$$\frac{802+112\sqrt{22}}{64\left(\sqrt{11}\right)^{2}-\left(-7\sqrt{2}\right)^{2}}$$

The square of \sqrt{11} is 11.

$$\frac{802+112\sqrt{22}}{64\times 11-\left(-7\sqrt{2}\right)^{2}}$$

Multiply 64 and 11 to get 704.

$$\frac{802+112\sqrt{22}}{704-\left(-7\sqrt{2}\right)^{2}}$$

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

$$\frac{802+112\sqrt{22}}{704-\left(-7\right)^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate -7 to the power of 2 and get 49.

$$\frac{802+112\sqrt{22}}{704-49\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{802+112\sqrt{22}}{704-49\times 2}$$

Multiply 49 and 2 to get 98.

$$\frac{802+112\sqrt{22}}{704-98}$$

Subtract 98 from 704 to get 606.

$$\frac{802+112\sqrt{22}}{606}$$