Solve 6/2-24/sqrt{20}+2sqrt{50}-3sqrt{8}

$$\frac { 6 } { 2 } - \frac { 24 } { \sqrt { 20 } } + 2 \sqrt { 50 } - 3 \sqrt { 8 }$$

Evaluate

$$-\frac{12\sqrt{5}}{5}+4\sqrt{2}+3\approx 3.290291103$$

Short Solution Steps

Divide 6 by 2 to get 3.

$$3-\frac{24}{\sqrt{20}}+2\sqrt{50}-3\sqrt{8}$$

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}.

$$3-\frac{24}{2\sqrt{5}}+2\sqrt{50}-3\sqrt{8}$$

Rationalize the denominator of \frac{24}{2\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$3-\frac{24\sqrt{5}}{2\left(\sqrt{5}\right)^{2}}+2\sqrt{50}-3\sqrt{8}$$

The square of \sqrt{5} is 5.

$$3-\frac{24\sqrt{5}}{2\times 5}+2\sqrt{50}-3\sqrt{8}$$

Cancel out 2 in both numerator and denominator.

$$3-\frac{12\sqrt{5}}{5}+2\sqrt{50}-3\sqrt{8}$$

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}.

$$3-\frac{12\sqrt{5}}{5}+2\times 5\sqrt{2}-3\sqrt{8}$$

Multiply 2 and 5 to get 10.

$$3-\frac{12\sqrt{5}}{5}+10\sqrt{2}-3\sqrt{8}$$

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}.

$$3-\frac{12\sqrt{5}}{5}+10\sqrt{2}-3\times 2\sqrt{2}$$

Multiply -3 and 2 to get -6.

$$3-\frac{12\sqrt{5}}{5}+10\sqrt{2}-6\sqrt{2}$$

Combine 10\sqrt{2} and -6\sqrt{2} to get 4\sqrt{2}.

$$3-\frac{12\sqrt{5}}{5}+4\sqrt{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3+4\sqrt{2} times \frac{5}{5}.

$$\frac{5\left(3+4\sqrt{2}\right)}{5}-\frac{12\sqrt{5}}{5}$$

Since \frac{5\left(3+4\sqrt{2}\right)}{5} and \frac{12\sqrt{5}}{5} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5\left(3+4\sqrt{2}\right)-12\sqrt{5}}{5}$$

Do the multiplications in 5\left(3+4\sqrt{2}\right)-12\sqrt{5}.

$$\frac{15+20\sqrt{2}-12\sqrt{5}}{5}$$