Solve 3/2sqrt{20}x(-sqrt{15})div(-1/3sqrt{48})+frac{sqrt{3}+sqrt{6}}{sqrt{3}}+sqrt{(1-sqrt{2})^2}

$$\frac { 3 } { 2 } \sqrt { 20 } x ( - \sqrt { 15 } ) \div ( - \frac { 1 } { 3 } \sqrt { 48 } ) + \frac { \sqrt { 3 } + \sqrt { 6 } } { \sqrt { 3 } } + \sqrt { ( 1 - \sqrt { 2 } ) ^ { 2 } }$$

Factor

$$\frac{45x+8\sqrt{2}}{4}$$

Solution Steps

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$factor(\frac{\frac{3}{2}\times 2\sqrt{5}x\left(-\sqrt{15}\right)}{-\frac{1}{3}\sqrt{48}}+\frac{\sqrt{3}+\sqrt{6}}{\sqrt{3}}+\sqrt{\left(1-\sqrt{2}\right)^{2}})$$

Multiply \frac{3}{2} and 2 to get 3.

$$factor(\frac{3\sqrt{5}x\left(-\sqrt{15}\right)}{-\frac{1}{3}\sqrt{48}}+\frac{\sqrt{3}+\sqrt{6}}{\sqrt{3}}+\sqrt{\left(1-\sqrt{2}\right)^{2}})$$

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

$$factor(\frac{3\sqrt{5}x\left(-\sqrt{15}\right)}{-\frac{1}{3}\times 4\sqrt{3}}+\frac{\sqrt{3}+\sqrt{6}}{\sqrt{3}}+\sqrt{\left(1-\sqrt{2}\right)^{2}})$$

Multiply -\frac{1}{3} and 4 to get -\frac{4}{3}.

$$factor(\frac{3\sqrt{5}x\left(-\sqrt{15}\right)}{-\frac{4}{3}\sqrt{3}}+\frac{\sqrt{3}+\sqrt{6}}{\sqrt{3}}+\sqrt{\left(1-\sqrt{2}\right)^{2}})$$

Rationalize the denominator of \frac{3\sqrt{5}x\left(-\sqrt{15}\right)}{-\frac{4}{3}\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$factor(\frac{3\sqrt{5}x\left(-\sqrt{15}\right)\sqrt{3}}{-\frac{4}{3}\left(\sqrt{3}\right)^{2}}+\frac{\sqrt{3}+\sqrt{6}}{\sqrt{3}}+\sqrt{\left(1-\sqrt{2}\right)^{2}})$$

The square of \sqrt{3} is 3.

$$factor(\frac{3\sqrt{5}x\left(-\sqrt{15}\right)\sqrt{3}}{-\frac{4}{3}\times 3}+\frac{\sqrt{3}+\sqrt{6}}{\sqrt{3}}+\sqrt{\left(1-\sqrt{2}\right)^{2}})$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$factor(\frac{3\sqrt{15}x\left(-\sqrt{15}\right)}{-\frac{4}{3}\times 3}+\frac{\sqrt{3}+\sqrt{6}}{\sqrt{3}}+\sqrt{\left(1-\sqrt{2}\right)^{2}})$$

Multiply -\frac{4}{3} and 3 to get -4.

$$factor(\frac{3\sqrt{15}x\left(-\sqrt{15}\right)}{-4}+\frac{\sqrt{3}+\sqrt{6}}{\sqrt{3}}+\sqrt{\left(1-\sqrt{2}\right)^{2}})$$

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

$$factor(\frac{3\sqrt{15}x\left(-\sqrt{15}\right)}{-4}+\frac{\left(\sqrt{3}+\sqrt{6}\right)\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\sqrt{\left(1-\sqrt{2}\right)^{2}})$$

The square of \sqrt{3} is 3.

$$factor(\frac{3\sqrt{15}x\left(-\sqrt{15}\right)}{-4}+\frac{\left(\sqrt{3}+\sqrt{6}\right)\sqrt{3}}{3}+\sqrt{\left(1-\sqrt{2}\right)^{2}})$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-\sqrt{2}\right)^{2}.

$$factor(\frac{3\sqrt{15}x\left(-\sqrt{15}\right)}{-4}+\frac{\left(\sqrt{3}+\sqrt{6}\right)\sqrt{3}}{3}+\sqrt{1-2\sqrt{2}+\left(\sqrt{2}\right)^{2}})$$

The square of \sqrt{2} is 2.

$$factor(\frac{3\sqrt{15}x\left(-\sqrt{15}\right)}{-4}+\frac{\left(\sqrt{3}+\sqrt{6}\right)\sqrt{3}}{3}+\sqrt{1-2\sqrt{2}+2})$$

Add 1 and 2 to get 3.

$$factor(\frac{3\sqrt{15}x\left(-\sqrt{15}\right)}{-4}+\frac{\left(\sqrt{3}+\sqrt{6}\right)\sqrt{3}}{3}+\sqrt{3-2\sqrt{2}})$$

Use the distributive property to multiply \sqrt{3}+\sqrt{6} by \sqrt{3}.

$$factor(\frac{3\sqrt{15}x\left(-\sqrt{15}\right)}{-4}+\frac{\left(\sqrt{3}\right)^{2}+\sqrt{6}\sqrt{3}}{3}+\sqrt{3-2\sqrt{2}})$$

The square of \sqrt{3} is 3.

$$factor(\frac{3\sqrt{15}x\left(-\sqrt{15}\right)}{-4}+\frac{3+\sqrt{6}\sqrt{3}}{3}+\sqrt{3-2\sqrt{2}})$$

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}.

$$factor(\frac{3\sqrt{15}x\left(-\sqrt{15}\right)}{-4}+\frac{3+\sqrt{3}\sqrt{2}\sqrt{3}}{3}+\sqrt{3-2\sqrt{2}})$$

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

$$factor(\frac{3\sqrt{15}x\left(-\sqrt{15}\right)}{-4}+\frac{3+3\sqrt{2}}{3}+\sqrt{3-2\sqrt{2}})$$

Divide each term of 3+3\sqrt{2} by 3 to get 1+\sqrt{2}.

$$factor(\frac{3\sqrt{15}x\left(-\sqrt{15}\right)}{-4}+1+\sqrt{2}+\sqrt{3-2\sqrt{2}})$$

Multiply 3 and -1 to get -3.

$$factor(\frac{-3\sqrt{15}x\sqrt{15}}{-4}+1+\sqrt{2}+\sqrt{3-2\sqrt{2}})$$

Multiply \sqrt{15} and \sqrt{15} to get 15.

$$factor(\frac{-3\times 15x}{-4}+1+\sqrt{2}+\sqrt{3-2\sqrt{2}})$$

Multiply -3 and 15 to get -45.

$$factor(\frac{-45x}{-4}+1+\sqrt{2}+\sqrt{3-2\sqrt{2}})$$

Factor out \frac{1}{4}.

$$\frac{45x+4+4\sqrt{2}+4\sqrt{3-2\sqrt{2}}}{4}$$

Evaluate

$$\frac{45x}{4}+2\sqrt{2}$$