Solve frac{4sqrt{2}-sqrt{11}}{3sqrt{2}+sqrt{11}}

$$\frac { 4 \sqrt { 2 } - \sqrt { 11 } } { 3 \sqrt { 2 } + \sqrt { 11 } }$$

Evaluate

$$5-\sqrt{22}\approx 0.30958424$$

Solution Steps

Rationalize the denominator of \frac{4\sqrt{2}-\sqrt{11}}{3\sqrt{2}+\sqrt{11}} by multiplying numerator and denominator by 3\sqrt{2}-\sqrt{11}.

$$\frac{\left(4\sqrt{2}-\sqrt{11}\right)\left(3\sqrt{2}-\sqrt{11}\right)}{\left(3\sqrt{2}+\sqrt{11}\right)\left(3\sqrt{2}-\sqrt{11}\right)}$$

Consider \left(3\sqrt{2}+\sqrt{11}\right)\left(3\sqrt{2}-\sqrt{11}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(4\sqrt{2}-\sqrt{11}\right)\left(3\sqrt{2}-\sqrt{11}\right)}{\left(3\sqrt{2}\right)^{2}-\left(\sqrt{11}\right)^{2}}$$

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

$$\frac{\left(4\sqrt{2}-\sqrt{11}\right)\left(3\sqrt{2}-\sqrt{11}\right)}{3^{2}\left(\sqrt{2}\right)^{2}-\left(\sqrt{11}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\left(4\sqrt{2}-\sqrt{11}\right)\left(3\sqrt{2}-\sqrt{11}\right)}{9\left(\sqrt{2}\right)^{2}-\left(\sqrt{11}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(4\sqrt{2}-\sqrt{11}\right)\left(3\sqrt{2}-\sqrt{11}\right)}{9\times 2-\left(\sqrt{11}\right)^{2}}$$

Multiply 9 and 2 to get 18.

$$\frac{\left(4\sqrt{2}-\sqrt{11}\right)\left(3\sqrt{2}-\sqrt{11}\right)}{18-\left(\sqrt{11}\right)^{2}}$$

The square of \sqrt{11} is 11.

$$\frac{\left(4\sqrt{2}-\sqrt{11}\right)\left(3\sqrt{2}-\sqrt{11}\right)}{18-11}$$

Subtract 11 from 18 to get 7.

$$\frac{\left(4\sqrt{2}-\sqrt{11}\right)\left(3\sqrt{2}-\sqrt{11}\right)}{7}$$

Apply the distributive property by multiplying each term of 4\sqrt{2}-\sqrt{11} by each term of 3\sqrt{2}-\sqrt{11}.

$$\frac{12\left(\sqrt{2}\right)^{2}-4\sqrt{11}\sqrt{2}-3\sqrt{11}\sqrt{2}+\left(\sqrt{11}\right)^{2}}{7}$$

The square of \sqrt{2} is 2.

$$\frac{12\times 2-4\sqrt{11}\sqrt{2}-3\sqrt{11}\sqrt{2}+\left(\sqrt{11}\right)^{2}}{7}$$

Multiply 12 and 2 to get 24.

$$\frac{24-4\sqrt{11}\sqrt{2}-3\sqrt{11}\sqrt{2}+\left(\sqrt{11}\right)^{2}}{7}$$

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

$$\frac{24-4\sqrt{22}-3\sqrt{11}\sqrt{2}+\left(\sqrt{11}\right)^{2}}{7}$$

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

$$\frac{24-4\sqrt{22}-3\sqrt{22}+\left(\sqrt{11}\right)^{2}}{7}$$

Combine -4\sqrt{22} and -3\sqrt{22} to get -7\sqrt{22}.

$$\frac{24-7\sqrt{22}+\left(\sqrt{11}\right)^{2}}{7}$$

The square of \sqrt{11} is 11.

$$\frac{24-7\sqrt{22}+11}{7}$$

Add 24 and 11 to get 35.

$$\frac{35-7\sqrt{22}}{7}$$

Divide each term of 35-7\sqrt{22} by 7 to get 5-\sqrt{22}.

$$5-\sqrt{22}$$