Solve frac{24-3sqrt{2}}{6sqrt{2}-11}

$$\frac{ 24-3 \sqrt{ 2 } }{ 6 \sqrt{ 2 } -11 }$$

Evaluate

$$\frac{-111\sqrt{2}-228}{49}\approx -7.856687866$$

Solution Steps

Rationalize the denominator of \frac{24-3\sqrt{2}}{6\sqrt{2}-11} by multiplying numerator and denominator by 6\sqrt{2}+11.

$$\frac{\left(24-3\sqrt{2}\right)\left(6\sqrt{2}+11\right)}{\left(6\sqrt{2}-11\right)\left(6\sqrt{2}+11\right)}$$

Consider \left(6\sqrt{2}-11\right)\left(6\sqrt{2}+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}.

$$\frac{\left(24-3\sqrt{2}\right)\left(6\sqrt{2}+11\right)}{\left(6\sqrt{2}\right)^{2}-11^{2}}$$

Expand \left(6\sqrt{2}\right)^{2}.

$$\frac{\left(24-3\sqrt{2}\right)\left(6\sqrt{2}+11\right)}{6^{2}\left(\sqrt{2}\right)^{2}-11^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{\left(24-3\sqrt{2}\right)\left(6\sqrt{2}+11\right)}{36\left(\sqrt{2}\right)^{2}-11^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(24-3\sqrt{2}\right)\left(6\sqrt{2}+11\right)}{36\times 2-11^{2}}$$

Multiply 36 and 2 to get 72.

$$\frac{\left(24-3\sqrt{2}\right)\left(6\sqrt{2}+11\right)}{72-11^{2}}$$

Calculate 11 to the power of 2 and get 121.

$$\frac{\left(24-3\sqrt{2}\right)\left(6\sqrt{2}+11\right)}{72-121}$$

Subtract 121 from 72 to get -49.

$$\frac{\left(24-3\sqrt{2}\right)\left(6\sqrt{2}+11\right)}{-49}$$

Apply the distributive property by multiplying each term of 24-3\sqrt{2} by each term of 6\sqrt{2}+11.

$$\frac{144\sqrt{2}+264-18\left(\sqrt{2}\right)^{2}-33\sqrt{2}}{-49}$$

The square of \sqrt{2} is 2.

$$\frac{144\sqrt{2}+264-18\times 2-33\sqrt{2}}{-49}$$

Multiply -18 and 2 to get -36.

$$\frac{144\sqrt{2}+264-36-33\sqrt{2}}{-49}$$

Subtract 36 from 264 to get 228.

$$\frac{144\sqrt{2}+228-33\sqrt{2}}{-49}$$

Combine 144\sqrt{2} and -33\sqrt{2} to get 111\sqrt{2}.

$$\frac{111\sqrt{2}+228}{-49}$$

Factor

$$\frac{3 {(-37 \sqrt{2} - 76)}}{49} = -7.85668786578395$$