Solve (sqrt{8}-2sqrt{0.25)}-(sqrt{11/8}+sqrt{50}+2/3sqrt{12})

$$( \sqrt { 8 } - 2 \sqrt { 0.25 ) } - ( \sqrt { 1 \frac { 1 } { 8 } } + \sqrt { 50 } + \frac { 2 } { 3 } \sqrt { 12 } )$$

Evaluate

$$-\frac{4\sqrt{3}}{3}-\frac{15\sqrt{2}}{4}-1\approx -8.612701936$$

Solution Steps

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}.

$$2\sqrt{2}-2\sqrt{0.25}-\left(\sqrt{\frac{1\times 8+1}{8}}+\sqrt{50}+\frac{2}{3}\sqrt{12}\right)$$

Calculate the square root of 0.25 and get 0.5.

$$2\sqrt{2}-2\times 0.5-\left(\sqrt{\frac{1\times 8+1}{8}}+\sqrt{50}+\frac{2}{3}\sqrt{12}\right)$$

Multiply -2 and 0.5 to get -1.

$$2\sqrt{2}-1-\left(\sqrt{\frac{1\times 8+1}{8}}+\sqrt{50}+\frac{2}{3}\sqrt{12}\right)$$

Multiply 1 and 8 to get 8.

$$2\sqrt{2}-1-\left(\sqrt{\frac{8+1}{8}}+\sqrt{50}+\frac{2}{3}\sqrt{12}\right)$$

Add 8 and 1 to get 9.

$$2\sqrt{2}-1-\left(\sqrt{\frac{9}{8}}+\sqrt{50}+\frac{2}{3}\sqrt{12}\right)$$

Rewrite the square root of the division \sqrt{\frac{9}{8}} as the division of square roots \frac{\sqrt{9}}{\sqrt{8}}.

$$2\sqrt{2}-1-\left(\frac{\sqrt{9}}{\sqrt{8}}+\sqrt{50}+\frac{2}{3}\sqrt{12}\right)$$

Calculate the square root of 9 and get 3.

$$2\sqrt{2}-1-\left(\frac{3}{\sqrt{8}}+\sqrt{50}+\frac{2}{3}\sqrt{12}\right)$$

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}.

$$2\sqrt{2}-1-\left(\frac{3}{2\sqrt{2}}+\sqrt{50}+\frac{2}{3}\sqrt{12}\right)$$

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

$$2\sqrt{2}-1-\left(\frac{3\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}+\sqrt{50}+\frac{2}{3}\sqrt{12}\right)$$

The square of \sqrt{2} is 2.

$$2\sqrt{2}-1-\left(\frac{3\sqrt{2}}{2\times 2}+\sqrt{50}+\frac{2}{3}\sqrt{12}\right)$$

Multiply 2 and 2 to get 4.

$$2\sqrt{2}-1-\left(\frac{3\sqrt{2}}{4}+\sqrt{50}+\frac{2}{3}\sqrt{12}\right)$$

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}.

$$2\sqrt{2}-1-\left(\frac{3\sqrt{2}}{4}+5\sqrt{2}+\frac{2}{3}\sqrt{12}\right)$$

Combine \frac{3\sqrt{2}}{4} and 5\sqrt{2} to get \frac{23}{4}\sqrt{2}.

$$2\sqrt{2}-1-\left(\frac{23}{4}\sqrt{2}+\frac{2}{3}\sqrt{12}\right)$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$2\sqrt{2}-1-\left(\frac{23}{4}\sqrt{2}+\frac{2}{3}\times 2\sqrt{3}\right)$$

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

$$2\sqrt{2}-1-\left(\frac{23}{4}\sqrt{2}+\frac{2\times 2}{3}\sqrt{3}\right)$$

Multiply 2 and 2 to get 4.

$$2\sqrt{2}-1-\left(\frac{23}{4}\sqrt{2}+\frac{4}{3}\sqrt{3}\right)$$

To find the opposite of \frac{23}{4}\sqrt{2}+\frac{4}{3}\sqrt{3}, find the opposite of each term.

$$2\sqrt{2}-1-\frac{23}{4}\sqrt{2}-\frac{4}{3}\sqrt{3}$$

Combine 2\sqrt{2} and -\frac{23}{4}\sqrt{2} to get -\frac{15}{4}\sqrt{2}.

$$-\frac{15}{4}\sqrt{2}-1-\frac{4}{3}\sqrt{3}$$

Factor

$$\frac{-16 \sqrt{3} - 45 \sqrt{2} - 12}{12} = -8.61270193565761$$