Solve sqrt{72}-4sqrt{1/2}-1/7sqrt{88}+sqrt{11/8}

$$\sqrt { 72 } - 4 \sqrt { \frac { 1 } { 2 } } - \frac { 1 } { 7 } \sqrt { 88 } + \sqrt { 1 \frac { 1 } { 8 } }$$

Evaluate

$$\frac{19\sqrt{2}}{4}-\frac{2\sqrt{22}}{7}\approx 5.377395633$$

Solution Steps

Factor 72=6^{2}\times 2. Rewrite the square root of the product \sqrt{6^{2}\times 2} as the product of square roots \sqrt{6^{2}}\sqrt{2}. Take the square root of 6^{2}.

$$6\sqrt{2}-4\sqrt{\frac{1}{2}}-\frac{1}{7}\sqrt{88}+\sqrt{\frac{1\times 8+1}{8}}$$

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

$$6\sqrt{2}-4\times \frac{\sqrt{1}}{\sqrt{2}}-\frac{1}{7}\sqrt{88}+\sqrt{\frac{1\times 8+1}{8}}$$

Calculate the square root of 1 and get 1.

$$6\sqrt{2}-4\times \frac{1}{\sqrt{2}}-\frac{1}{7}\sqrt{88}+\sqrt{\frac{1\times 8+1}{8}}$$

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$6\sqrt{2}-4\times \frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-\frac{1}{7}\sqrt{88}+\sqrt{\frac{1\times 8+1}{8}}$$

The square of \sqrt{2} is 2.

$$6\sqrt{2}-4\times \frac{\sqrt{2}}{2}-\frac{1}{7}\sqrt{88}+\sqrt{\frac{1\times 8+1}{8}}$$

Cancel out 2, the greatest common factor in 4 and 2.

$$6\sqrt{2}-2\sqrt{2}-\frac{1}{7}\sqrt{88}+\sqrt{\frac{1\times 8+1}{8}}$$

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

$$4\sqrt{2}-\frac{1}{7}\sqrt{88}+\sqrt{\frac{1\times 8+1}{8}}$$

Factor 88=2^{2}\times 22. Rewrite the square root of the product \sqrt{2^{2}\times 22} as the product of square roots \sqrt{2^{2}}\sqrt{22}. Take the square root of 2^{2}.

$$4\sqrt{2}-\frac{1}{7}\times 2\sqrt{22}+\sqrt{\frac{1\times 8+1}{8}}$$

Express -\frac{1}{7}\times 2 as a single fraction.

$$4\sqrt{2}+\frac{-2}{7}\sqrt{22}+\sqrt{\frac{1\times 8+1}{8}}$$

Fraction \frac{-2}{7} can be rewritten as -\frac{2}{7} by extracting the negative sign.

$$4\sqrt{2}-\frac{2}{7}\sqrt{22}+\sqrt{\frac{1\times 8+1}{8}}$$

Multiply 1 and 8 to get 8.

$$4\sqrt{2}-\frac{2}{7}\sqrt{22}+\sqrt{\frac{8+1}{8}}$$

Add 8 and 1 to get 9.

$$4\sqrt{2}-\frac{2}{7}\sqrt{22}+\sqrt{\frac{9}{8}}$$

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

$$4\sqrt{2}-\frac{2}{7}\sqrt{22}+\frac{\sqrt{9}}{\sqrt{8}}$$

Calculate the square root of 9 and get 3.

$$4\sqrt{2}-\frac{2}{7}\sqrt{22}+\frac{3}{\sqrt{8}}$$

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}.

$$4\sqrt{2}-\frac{2}{7}\sqrt{22}+\frac{3}{2\sqrt{2}}$$

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

$$4\sqrt{2}-\frac{2}{7}\sqrt{22}+\frac{3\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$4\sqrt{2}-\frac{2}{7}\sqrt{22}+\frac{3\sqrt{2}}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$4\sqrt{2}-\frac{2}{7}\sqrt{22}+\frac{3\sqrt{2}}{4}$$

Combine 4\sqrt{2} and \frac{3\sqrt{2}}{4} to get \frac{19}{4}\sqrt{2}.

$$\frac{19}{4}\sqrt{2}-\frac{2}{7}\sqrt{22}$$

Factor

$$\frac{133 \sqrt{2} - 8 \sqrt{22}}{28} = 5.377395632751222$$