Solve sqrt{(8div1/2)(1/2+8)+2}-sqrt{(8div1/2)(1/2+8)-2}

$$\sqrt { ( 8 \div \frac { 1 } { 2 } ) ( \frac { 1 } { 2 } + 8 ) + 2 } - \sqrt { ( 8 \div \frac { 1 } { 2 } ) ( \frac { 1 } { 2 } + 8 ) - 2 }$$

Evaluate

$$\sqrt{138}-\sqrt{134}\approx 0.171503222$$

Solution Steps

Divide 8 by \frac{1}{2} by multiplying 8 by the reciprocal of \frac{1}{2}.

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

Multiply 8 and 2 to get 16.

$$\sqrt{16\left(\frac{1}{2}+8\right)+2}-\sqrt{\frac{8}{\frac{1}{2}}\left(\frac{1}{2}+8\right)-2}$$

Convert 8 to fraction \frac{16}{2}.

$$\sqrt{16\left(\frac{1}{2}+\frac{16}{2}\right)+2}-\sqrt{\frac{8}{\frac{1}{2}}\left(\frac{1}{2}+8\right)-2}$$

Since \frac{1}{2} and \frac{16}{2} have the same denominator, add them by adding their numerators.

$$\sqrt{16\times \frac{1+16}{2}+2}-\sqrt{\frac{8}{\frac{1}{2}}\left(\frac{1}{2}+8\right)-2}$$

Add 1 and 16 to get 17.

$$\sqrt{16\times \frac{17}{2}+2}-\sqrt{\frac{8}{\frac{1}{2}}\left(\frac{1}{2}+8\right)-2}$$

Express 16\times \frac{17}{2} as a single fraction.

$$\sqrt{\frac{16\times 17}{2}+2}-\sqrt{\frac{8}{\frac{1}{2}}\left(\frac{1}{2}+8\right)-2}$$

Multiply 16 and 17 to get 272.

$$\sqrt{\frac{272}{2}+2}-\sqrt{\frac{8}{\frac{1}{2}}\left(\frac{1}{2}+8\right)-2}$$

Divide 272 by 2 to get 136.

$$\sqrt{136+2}-\sqrt{\frac{8}{\frac{1}{2}}\left(\frac{1}{2}+8\right)-2}$$

Add 136 and 2 to get 138.

$$\sqrt{138}-\sqrt{\frac{8}{\frac{1}{2}}\left(\frac{1}{2}+8\right)-2}$$

Divide 8 by \frac{1}{2} by multiplying 8 by the reciprocal of \frac{1}{2}.

$$\sqrt{138}-\sqrt{8\times 2\left(\frac{1}{2}+8\right)-2}$$

Multiply 8 and 2 to get 16.

$$\sqrt{138}-\sqrt{16\left(\frac{1}{2}+8\right)-2}$$

Convert 8 to fraction \frac{16}{2}.

$$\sqrt{138}-\sqrt{16\left(\frac{1}{2}+\frac{16}{2}\right)-2}$$

Since \frac{1}{2} and \frac{16}{2} have the same denominator, add them by adding their numerators.

$$\sqrt{138}-\sqrt{16\times \frac{1+16}{2}-2}$$

Add 1 and 16 to get 17.

$$\sqrt{138}-\sqrt{16\times \frac{17}{2}-2}$$

Express 16\times \frac{17}{2} as a single fraction.

$$\sqrt{138}-\sqrt{\frac{16\times 17}{2}-2}$$

Multiply 16 and 17 to get 272.

$$\sqrt{138}-\sqrt{\frac{272}{2}-2}$$

Divide 272 by 2 to get 136.

$$\sqrt{138}-\sqrt{136-2}$$

Subtract 2 from 136 to get 134.

$$\sqrt{138}-\sqrt{134}$$