Solve frac{6(sqrt{3}+1)}{6sqrt{6}+6sqrt{2}}

$$\frac{ 6( \sqrt{ 3 } +1) }{ 6 \sqrt{ 6 } +6 \sqrt{ 2 } }$$

Evaluate

$$\frac{\sqrt{2}}{2}\approx 0.707106781$$

Solution Steps

Rationalize the denominator of \frac{6\left(\sqrt{3}+1\right)}{6\sqrt{6}+6\sqrt{2}} by multiplying numerator and denominator by 6\sqrt{6}-6\sqrt{2}.

$$\frac{6\left(\sqrt{3}+1\right)\left(6\sqrt{6}-6\sqrt{2}\right)}{\left(6\sqrt{6}+6\sqrt{2}\right)\left(6\sqrt{6}-6\sqrt{2}\right)}$$

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

$$\frac{6\left(\sqrt{3}+1\right)\left(6\sqrt{6}-6\sqrt{2}\right)}{\left(6\sqrt{6}\right)^{2}-\left(6\sqrt{2}\right)^{2}}$$

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

$$\frac{6\left(\sqrt{3}+1\right)\left(6\sqrt{6}-6\sqrt{2}\right)}{6^{2}\left(\sqrt{6}\right)^{2}-\left(6\sqrt{2}\right)^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{6\left(\sqrt{3}+1\right)\left(6\sqrt{6}-6\sqrt{2}\right)}{36\left(\sqrt{6}\right)^{2}-\left(6\sqrt{2}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{6\left(\sqrt{3}+1\right)\left(6\sqrt{6}-6\sqrt{2}\right)}{36\times 6-\left(6\sqrt{2}\right)^{2}}$$

Multiply 36 and 6 to get 216.

$$\frac{6\left(\sqrt{3}+1\right)\left(6\sqrt{6}-6\sqrt{2}\right)}{216-\left(6\sqrt{2}\right)^{2}}$$

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

$$\frac{6\left(\sqrt{3}+1\right)\left(6\sqrt{6}-6\sqrt{2}\right)}{216-6^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{6\left(\sqrt{3}+1\right)\left(6\sqrt{6}-6\sqrt{2}\right)}{216-36\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{6\left(\sqrt{3}+1\right)\left(6\sqrt{6}-6\sqrt{2}\right)}{216-36\times 2}$$

Multiply 36 and 2 to get 72.

$$\frac{6\left(\sqrt{3}+1\right)\left(6\sqrt{6}-6\sqrt{2}\right)}{216-72}$$

Subtract 72 from 216 to get 144.

$$\frac{6\left(\sqrt{3}+1\right)\left(6\sqrt{6}-6\sqrt{2}\right)}{144}$$

Divide 6\left(\sqrt{3}+1\right)\left(6\sqrt{6}-6\sqrt{2}\right) by 144 to get \frac{1}{24}\left(\sqrt{3}+1\right)\left(6\sqrt{6}-6\sqrt{2}\right).

$$\frac{1}{24}\left(\sqrt{3}+1\right)\left(6\sqrt{6}-6\sqrt{2}\right)$$

Use the distributive property to multiply \frac{1}{24} by \sqrt{3}+1.

$$\left(\frac{1}{24}\sqrt{3}+\frac{1}{24}\right)\left(6\sqrt{6}-6\sqrt{2}\right)$$

Apply the distributive property by multiplying each term of \frac{1}{24}\sqrt{3}+\frac{1}{24} by each term of 6\sqrt{6}-6\sqrt{2}.

$$\frac{1}{24}\sqrt{3}\times 6\sqrt{6}+\frac{1}{24}\sqrt{3}\left(-6\right)\sqrt{2}+\frac{1}{24}\times 6\sqrt{6}+\frac{1}{24}\left(-6\right)\sqrt{2}$$

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}.

$$\frac{1}{24}\sqrt{3}\times 6\sqrt{3}\sqrt{2}+\frac{1}{24}\sqrt{3}\left(-6\right)\sqrt{2}+\frac{1}{24}\times 6\sqrt{6}+\frac{1}{24}\left(-6\right)\sqrt{2}$$

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

$$\frac{1}{24}\times 3\times 6\sqrt{2}+\frac{1}{24}\sqrt{3}\left(-6\right)\sqrt{2}+\frac{1}{24}\times 6\sqrt{6}+\frac{1}{24}\left(-6\right)\sqrt{2}$$

Multiply \frac{1}{24} and 3 to get \frac{3}{24}.

$$\frac{3}{24}\times 6\sqrt{2}+\frac{1}{24}\sqrt{3}\left(-6\right)\sqrt{2}+\frac{1}{24}\times 6\sqrt{6}+\frac{1}{24}\left(-6\right)\sqrt{2}$$

Reduce the fraction \frac{3}{24} to lowest terms by extracting and canceling out 3.

$$\frac{1}{8}\times 6\sqrt{2}+\frac{1}{24}\sqrt{3}\left(-6\right)\sqrt{2}+\frac{1}{24}\times 6\sqrt{6}+\frac{1}{24}\left(-6\right)\sqrt{2}$$

Multiply \frac{1}{8} and 6 to get \frac{6}{8}.

$$\frac{6}{8}\sqrt{2}+\frac{1}{24}\sqrt{3}\left(-6\right)\sqrt{2}+\frac{1}{24}\times 6\sqrt{6}+\frac{1}{24}\left(-6\right)\sqrt{2}$$

Reduce the fraction \frac{6}{8} to lowest terms by extracting and canceling out 2.

$$\frac{3}{4}\sqrt{2}+\frac{1}{24}\sqrt{3}\left(-6\right)\sqrt{2}+\frac{1}{24}\times 6\sqrt{6}+\frac{1}{24}\left(-6\right)\sqrt{2}$$

Multiply \frac{1}{24} and -6 to get \frac{-6}{24}.

$$\frac{3}{4}\sqrt{2}+\frac{-6}{24}\sqrt{3}\sqrt{2}+\frac{1}{24}\times 6\sqrt{6}+\frac{1}{24}\left(-6\right)\sqrt{2}$$

Reduce the fraction \frac{-6}{24} to lowest terms by extracting and canceling out 6.

$$\frac{3}{4}\sqrt{2}-\frac{1}{4}\sqrt{3}\sqrt{2}+\frac{1}{24}\times 6\sqrt{6}+\frac{1}{24}\left(-6\right)\sqrt{2}$$

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

$$\frac{3}{4}\sqrt{2}-\frac{1}{4}\sqrt{6}+\frac{1}{24}\times 6\sqrt{6}+\frac{1}{24}\left(-6\right)\sqrt{2}$$

Multiply \frac{1}{24} and 6 to get \frac{6}{24}.

$$\frac{3}{4}\sqrt{2}-\frac{1}{4}\sqrt{6}+\frac{6}{24}\sqrt{6}+\frac{1}{24}\left(-6\right)\sqrt{2}$$

Reduce the fraction \frac{6}{24} to lowest terms by extracting and canceling out 6.

$$\frac{3}{4}\sqrt{2}-\frac{1}{4}\sqrt{6}+\frac{1}{4}\sqrt{6}+\frac{1}{24}\left(-6\right)\sqrt{2}$$

Combine -\frac{1}{4}\sqrt{6} and \frac{1}{4}\sqrt{6} to get 0.

$$\frac{3}{4}\sqrt{2}+\frac{1}{24}\left(-6\right)\sqrt{2}$$

Multiply \frac{1}{24} and -6 to get \frac{-6}{24}.

$$\frac{3}{4}\sqrt{2}+\frac{-6}{24}\sqrt{2}$$

Reduce the fraction \frac{-6}{24} to lowest terms by extracting and canceling out 6.

$$\frac{3}{4}\sqrt{2}-\frac{1}{4}\sqrt{2}$$

Combine \frac{3}{4}\sqrt{2} and -\frac{1}{4}\sqrt{2} to get \frac{1}{2}\sqrt{2}.

$$\frac{1}{2}\sqrt{2}$$