Solve frac{sqrt{4}abc+sqrt{10}a^{2}b^{3}c^3-sqrt{6}a^3b^2c^6}{sqrt{2}abc}

$$\frac { \sqrt { 4 } a b c + \sqrt { 10 } a ^ { 2 } b ^ { 3 } c ^ { 3 } - \sqrt { 6 } a ^ { 3 } b ^ { 2 } c ^ { 6 } } { \sqrt { 2 } a b c }$$

Factor

$$\frac{\sqrt{2}\left(-\sqrt{6}ba^{2}c^{5}+\sqrt{10}ab^{2}c^{2}+2\right)}{2}$$

Solution Steps

Calculate the square root of 4 and get 2.

$$factor(\frac{2abc+\sqrt{10}a^{2}b^{3}c^{3}-\sqrt{6}a^{3}b^{2}c^{6}}{\sqrt{2}abc})$$

Factor the expressions that are not already factored in \frac{2abc+\sqrt{10}a^{2}b^{3}c^{3}-\sqrt{6}a^{3}b^{2}c^{6}}{\sqrt{2}abc}.

$$factor(\frac{abc\left(-\sqrt{6}ba^{2}c^{5}+\sqrt{10}ab^{2}c^{2}+2\right)}{\sqrt{2}abc})$$

Cancel out abc in both numerator and denominator.

$$factor(\frac{-\sqrt{6}ba^{2}c^{5}+\sqrt{10}ab^{2}c^{2}+2}{\sqrt{2}})$$

Rationalize the denominator of \frac{-\sqrt{6}ba^{2}c^{5}+\sqrt{10}ab^{2}c^{2}+2}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$factor(\frac{\left(-\sqrt{6}ba^{2}c^{5}+\sqrt{10}ab^{2}c^{2}+2\right)\sqrt{2}}{\left(\sqrt{2}\right)^{2}})$$

The square of \sqrt{2} is 2.

$$factor(\frac{\left(-\sqrt{6}ba^{2}c^{5}+\sqrt{10}ab^{2}c^{2}+2\right)\sqrt{2}}{2})$$

Use the distributive property to multiply -\sqrt{6}ba^{2}c^{5}+\sqrt{10}ab^{2}c^{2}+2 by \sqrt{2}.

$$factor(\frac{-\sqrt{6}ba^{2}c^{5}\sqrt{2}+\sqrt{10}ab^{2}c^{2}\sqrt{2}+2\sqrt{2}}{2})$$

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}.

$$factor(\frac{-\sqrt{2}\sqrt{3}ba^{2}c^{5}\sqrt{2}+\sqrt{10}ab^{2}c^{2}\sqrt{2}+2\sqrt{2}}{2})$$

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

$$factor(\frac{-2ba^{2}c^{5}\sqrt{3}+\sqrt{10}ab^{2}c^{2}\sqrt{2}+2\sqrt{2}}{2})$$

Factor 10=2\times 5. Rewrite the square root of the product \sqrt{2\times 5} as the product of square roots \sqrt{2}\sqrt{5}.

$$factor(\frac{-2ba^{2}c^{5}\sqrt{3}+\sqrt{2}\sqrt{5}ab^{2}c^{2}\sqrt{2}+2\sqrt{2}}{2})$$

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

$$factor(\frac{-2ba^{2}c^{5}\sqrt{3}+2ab^{2}c^{2}\sqrt{5}+2\sqrt{2}}{2})$$

Consider -2ba^{2}c^{5}\times 3^{\frac{1}{2}}+2ab^{2}c^{2}\times 5^{\frac{1}{2}}+2\times 2^{\frac{1}{2}}. Factor out 2.

$$2\left(-ba^{2}c^{5}\sqrt{3}+ab^{2}c^{2}\sqrt{5}+\sqrt{2}\right)$$

Rewrite the complete factored expression. Simplify.

$$-ba^{2}c^{5}\sqrt{3}+ab^{2}c^{2}\sqrt{5}+\sqrt{2}$$

Evaluate

$$-\frac{2\sqrt{3}ba^{2}c^{5}}{2}+\frac{2\sqrt{5}a\left(bc\right)^{2}}{2}+\sqrt{2}$$