Solve frac{5left(3+sqrt{3}right)frac{sqrt{2}}{2}}{frac{sqrt{6}+sqrt{2}}{4}}

$$\frac{ 5 \left( 3+ \sqrt{ 3 } \right) \frac{ \sqrt{ 2 } }{ 2 } }{ \frac{ \sqrt{ 6 } + \sqrt{ 2 } }{ 4 } }$$

Evaluate

$$10\sqrt{3}\approx 17.320508076$$

Solution Steps

Express 5\times \frac{\sqrt{2}}{2} as a single fraction.

$$\frac{\frac{5\sqrt{2}}{2}\left(3+\sqrt{3}\right)}{\frac{\sqrt{6}+\sqrt{2}}{4}}$$

Express \frac{5\sqrt{2}}{2}\left(3+\sqrt{3}\right) as a single fraction.

$$\frac{\frac{5\sqrt{2}\left(3+\sqrt{3}\right)}{2}}{\frac{\sqrt{6}+\sqrt{2}}{4}}$$

Divide \frac{5\sqrt{2}\left(3+\sqrt{3}\right)}{2} by \frac{\sqrt{6}+\sqrt{2}}{4} by multiplying \frac{5\sqrt{2}\left(3+\sqrt{3}\right)}{2} by the reciprocal of \frac{\sqrt{6}+\sqrt{2}}{4}.

$$\frac{5\sqrt{2}\left(3+\sqrt{3}\right)\times 4}{2\left(\sqrt{6}+\sqrt{2}\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{2\times 5\sqrt{2}\left(\sqrt{3}+3\right)}{\sqrt{2}+\sqrt{6}}$$

Rationalize the denominator of \frac{2\times 5\sqrt{2}\left(\sqrt{3}+3\right)}{\sqrt{2}+\sqrt{6}} by multiplying numerator and denominator by \sqrt{2}-\sqrt{6}.

$$\frac{2\times 5\sqrt{2}\left(\sqrt{3}+3\right)\left(\sqrt{2}-\sqrt{6}\right)}{\left(\sqrt{2}+\sqrt{6}\right)\left(\sqrt{2}-\sqrt{6}\right)}$$

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

$$\frac{2\times 5\sqrt{2}\left(\sqrt{3}+3\right)\left(\sqrt{2}-\sqrt{6}\right)}{\left(\sqrt{2}\right)^{2}-\left(\sqrt{6}\right)^{2}}$$

Square \sqrt{2}. Square \sqrt{6}.

$$\frac{2\times 5\sqrt{2}\left(\sqrt{3}+3\right)\left(\sqrt{2}-\sqrt{6}\right)}{2-6}$$

Subtract 6 from 2 to get -4.

$$\frac{2\times 5\sqrt{2}\left(\sqrt{3}+3\right)\left(\sqrt{2}-\sqrt{6}\right)}{-4}$$

Multiply 2 and 5 to get 10.

$$\frac{10\sqrt{2}\left(\sqrt{3}+3\right)\left(\sqrt{2}-\sqrt{6}\right)}{-4}$$

Divide 10\sqrt{2}\left(\sqrt{3}+3\right)\left(\sqrt{2}-\sqrt{6}\right) by -4 to get -\frac{5}{2}\sqrt{2}\left(\sqrt{3}+3\right)\left(\sqrt{2}-\sqrt{6}\right).

$$-\frac{5}{2}\sqrt{2}\left(\sqrt{3}+3\right)\left(\sqrt{2}-\sqrt{6}\right)$$

Use the distributive property to multiply -\frac{5}{2}\sqrt{2} by \sqrt{3}+3.

$$\left(-\frac{5}{2}\sqrt{2}\sqrt{3}-\frac{5}{2}\sqrt{2}\times 3\right)\left(\sqrt{2}-\sqrt{6}\right)$$

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

$$\left(-\frac{5}{2}\sqrt{6}-\frac{5}{2}\sqrt{2}\times 3\right)\left(\sqrt{2}-\sqrt{6}\right)$$

Express -\frac{5}{2}\times 3 as a single fraction.

$$\left(-\frac{5}{2}\sqrt{6}+\frac{-5\times 3}{2}\sqrt{2}\right)\left(\sqrt{2}-\sqrt{6}\right)$$

Multiply -5 and 3 to get -15.

$$\left(-\frac{5}{2}\sqrt{6}+\frac{-15}{2}\sqrt{2}\right)\left(\sqrt{2}-\sqrt{6}\right)$$

Fraction \frac{-15}{2} can be rewritten as -\frac{15}{2} by extracting the negative sign.

$$\left(-\frac{5}{2}\sqrt{6}-\frac{15}{2}\sqrt{2}\right)\left(\sqrt{2}-\sqrt{6}\right)$$

Apply the distributive property by multiplying each term of -\frac{5}{2}\sqrt{6}-\frac{15}{2}\sqrt{2} by each term of \sqrt{2}-\sqrt{6}.

$$-\frac{5}{2}\sqrt{6}\sqrt{2}-\frac{5}{2}\sqrt{6}\left(-1\right)\sqrt{6}-\frac{15}{2}\sqrt{2}\sqrt{2}-\frac{15}{2}\sqrt{2}\left(-1\right)\sqrt{6}$$

Multiply \sqrt{6} and \sqrt{6} to get 6.

$$-\frac{5}{2}\sqrt{6}\sqrt{2}-\frac{5}{2}\times 6\left(-1\right)-\frac{15}{2}\sqrt{2}\sqrt{2}-\frac{15}{2}\sqrt{2}\left(-1\right)\sqrt{6}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$-\frac{5}{2}\sqrt{6}\sqrt{2}-\frac{5}{2}\times 6\left(-1\right)-\frac{15}{2}\times 2-\frac{15}{2}\sqrt{2}\left(-1\right)\sqrt{6}$$

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

$$-\frac{5}{2}\sqrt{2}\sqrt{3}\sqrt{2}-\frac{5}{2}\times 6\left(-1\right)-\frac{15}{2}\times 2-\frac{15}{2}\sqrt{2}\left(-1\right)\sqrt{6}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$-\frac{5}{2}\times 2\sqrt{3}-\frac{5}{2}\times 6\left(-1\right)-\frac{15}{2}\times 2-\frac{15}{2}\sqrt{2}\left(-1\right)\sqrt{6}$$

Cancel out 2 and 2.

$$-5\sqrt{3}-\frac{5}{2}\times 6\left(-1\right)-\frac{15}{2}\times 2-\frac{15}{2}\sqrt{2}\left(-1\right)\sqrt{6}$$

Express -\frac{5}{2}\times 6 as a single fraction.

$$-5\sqrt{3}+\frac{-5\times 6}{2}\left(-1\right)-\frac{15}{2}\times 2-\frac{15}{2}\sqrt{2}\left(-1\right)\sqrt{6}$$

Multiply -5 and 6 to get -30.

$$-5\sqrt{3}+\frac{-30}{2}\left(-1\right)-\frac{15}{2}\times 2-\frac{15}{2}\sqrt{2}\left(-1\right)\sqrt{6}$$

Divide -30 by 2 to get -15.

$$-5\sqrt{3}-15\left(-1\right)-\frac{15}{2}\times 2-\frac{15}{2}\sqrt{2}\left(-1\right)\sqrt{6}$$

Multiply -15 and -1 to get 15.

$$-5\sqrt{3}+15-\frac{15}{2}\times 2-\frac{15}{2}\sqrt{2}\left(-1\right)\sqrt{6}$$

Cancel out 2 and 2.

$$-5\sqrt{3}+15-15-\frac{15}{2}\sqrt{2}\left(-1\right)\sqrt{6}$$

Subtract 15 from 15 to get 0.

$$-5\sqrt{3}-\frac{15}{2}\sqrt{2}\left(-1\right)\sqrt{6}$$

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

$$-5\sqrt{3}-\frac{15}{2}\sqrt{2}\left(-1\right)\sqrt{2}\sqrt{3}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$-5\sqrt{3}-\frac{15}{2}\times 2\left(-1\right)\sqrt{3}$$

Cancel out 2 and 2.

$$-5\sqrt{3}-15\left(-1\right)\sqrt{3}$$

Multiply -15 and -1 to get 15.

$$-5\sqrt{3}+15\sqrt{3}$$

Combine -5\sqrt{3} and 15\sqrt{3} to get 10\sqrt{3}.

$$10\sqrt{3}$$