Solve sqrt{50}-4/sqrt{8}+12/sqrt{18}-2sqrt{32}

$$\sqrt { 50 } - \frac { 4 } { \sqrt { 8 } } + \frac { 12 } { \sqrt { 18 } } - 2 \sqrt { 32 }$$

Evaluate

$$-2\sqrt{2}\approx -2.828427125$$

Solution Steps

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

$$5\sqrt{2}-\frac{4}{\sqrt{8}}+\frac{12}{\sqrt{18}}-2\sqrt{32}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$5\sqrt{2}-\frac{4}{2\sqrt{2}}+\frac{12}{\sqrt{18}}-2\sqrt{32}$$

Rationalize the denominator of \frac{4}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$5\sqrt{2}-\frac{4\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}+\frac{12}{\sqrt{18}}-2\sqrt{32}$$

The square of \sqrt{2} is 2.

$$5\sqrt{2}-\frac{4\sqrt{2}}{2\times 2}+\frac{12}{\sqrt{18}}-2\sqrt{32}$$

Cancel out 2\times 2 in both numerator and denominator.

$$5\sqrt{2}-\sqrt{2}+\frac{12}{\sqrt{18}}-2\sqrt{32}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$5\sqrt{2}-\sqrt{2}+\frac{12}{3\sqrt{2}}-2\sqrt{32}$$

Rationalize the denominator of \frac{12}{3\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$5\sqrt{2}-\sqrt{2}+\frac{12\sqrt{2}}{3\left(\sqrt{2}\right)^{2}}-2\sqrt{32}$$

The square of \sqrt{2} is 2.

$$5\sqrt{2}-\sqrt{2}+\frac{12\sqrt{2}}{3\times 2}-2\sqrt{32}$$

Cancel out 2\times 3 in both numerator and denominator.

$$5\sqrt{2}-\sqrt{2}+2\sqrt{2}-2\sqrt{32}$$

Combine 5\sqrt{2} and 2\sqrt{2} to get 7\sqrt{2}.

$$7\sqrt{2}-\sqrt{2}-2\sqrt{32}$$

Factor 32=4^{2}\times 2. Rewrite the square root of the product \sqrt{4^{2}\times 2} as the product of square roots \sqrt{4^{2}}\sqrt{2}. Take the square root of 4^{2}.

$$7\sqrt{2}-\sqrt{2}-2\times 4\sqrt{2}$$

Multiply -2 and 4 to get -8.

$$7\sqrt{2}-\sqrt{2}-8\sqrt{2}$$

Combine 7\sqrt{2} and -8\sqrt{2} to get -\sqrt{2}.

$$-\sqrt{2}-\sqrt{2}$$

Combine -\sqrt{2} and -\sqrt{2} to get -2\sqrt{2}.

$$-2\sqrt{2}$$