Solve frac{-13-9sqrt{3}}{32sqrt{2}+18sqrt{6}}

$$\frac { - 13 - 9 \sqrt { 3 } } { 32 \sqrt { 2 } + 18 \sqrt { 6 } }$$

Evaluate

$$\frac{35\sqrt{2}-27\sqrt{6}}{52}\approx -0.31997593$$

Solution Steps

Rationalize the denominator of \frac{-13-9\sqrt{3}}{32\sqrt{2}+18\sqrt{6}} by multiplying numerator and denominator by 32\sqrt{2}-18\sqrt{6}.

$$\frac{\left(-13-9\sqrt{3}\right)\left(32\sqrt{2}-18\sqrt{6}\right)}{\left(32\sqrt{2}+18\sqrt{6}\right)\left(32\sqrt{2}-18\sqrt{6}\right)}$$

Consider \left(32\sqrt{2}+18\sqrt{6}\right)\left(32\sqrt{2}-18\sqrt{6}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(-13-9\sqrt{3}\right)\left(32\sqrt{2}-18\sqrt{6}\right)}{\left(32\sqrt{2}\right)^{2}-\left(18\sqrt{6}\right)^{2}}$$

Expand \left(32\sqrt{2}\right)^{2}.

$$\frac{\left(-13-9\sqrt{3}\right)\left(32\sqrt{2}-18\sqrt{6}\right)}{32^{2}\left(\sqrt{2}\right)^{2}-\left(18\sqrt{6}\right)^{2}}$$

Calculate 32 to the power of 2 and get 1024.

$$\frac{\left(-13-9\sqrt{3}\right)\left(32\sqrt{2}-18\sqrt{6}\right)}{1024\left(\sqrt{2}\right)^{2}-\left(18\sqrt{6}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(-13-9\sqrt{3}\right)\left(32\sqrt{2}-18\sqrt{6}\right)}{1024\times 2-\left(18\sqrt{6}\right)^{2}}$$

Multiply 1024 and 2 to get 2048.

$$\frac{\left(-13-9\sqrt{3}\right)\left(32\sqrt{2}-18\sqrt{6}\right)}{2048-\left(18\sqrt{6}\right)^{2}}$$

Expand \left(18\sqrt{6}\right)^{2}.

$$\frac{\left(-13-9\sqrt{3}\right)\left(32\sqrt{2}-18\sqrt{6}\right)}{2048-18^{2}\left(\sqrt{6}\right)^{2}}$$

Calculate 18 to the power of 2 and get 324.

$$\frac{\left(-13-9\sqrt{3}\right)\left(32\sqrt{2}-18\sqrt{6}\right)}{2048-324\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{\left(-13-9\sqrt{3}\right)\left(32\sqrt{2}-18\sqrt{6}\right)}{2048-324\times 6}$$

Multiply 324 and 6 to get 1944.

$$\frac{\left(-13-9\sqrt{3}\right)\left(32\sqrt{2}-18\sqrt{6}\right)}{2048-1944}$$

Subtract 1944 from 2048 to get 104.

$$\frac{\left(-13-9\sqrt{3}\right)\left(32\sqrt{2}-18\sqrt{6}\right)}{104}$$

Apply the distributive property by multiplying each term of -13-9\sqrt{3} by each term of 32\sqrt{2}-18\sqrt{6}.

$$\frac{-416\sqrt{2}+234\sqrt{6}-288\sqrt{3}\sqrt{2}+162\sqrt{3}\sqrt{6}}{104}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{-416\sqrt{2}+234\sqrt{6}-288\sqrt{6}+162\sqrt{3}\sqrt{6}}{104}$$

Combine 234\sqrt{6} and -288\sqrt{6} to get -54\sqrt{6}.

$$\frac{-416\sqrt{2}-54\sqrt{6}+162\sqrt{3}\sqrt{6}}{104}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$\frac{-416\sqrt{2}-54\sqrt{6}+162\sqrt{3}\sqrt{3}\sqrt{2}}{104}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{-416\sqrt{2}-54\sqrt{6}+162\times 3\sqrt{2}}{104}$$

Multiply 162 and 3 to get 486.

$$\frac{-416\sqrt{2}-54\sqrt{6}+486\sqrt{2}}{104}$$

Combine -416\sqrt{2} and 486\sqrt{2} to get 70\sqrt{2}.

$$\frac{70\sqrt{2}-54\sqrt{6}}{104}$$