Solve 1/sqrt{3+1}-frac{sqrt{3}}{2}+sqrt{32}*sqrt{1/8}

$$\frac { 1 } { \sqrt { 3 } + 1 } - \frac { \sqrt { 3 } } { 2 } + \sqrt { 32 } \times \sqrt { \frac { 1 } { 8 } }$$

Evaluate

$$\frac{3}{2}=1.5$$

Short Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{3}+1} by multiplying numerator and denominator by \sqrt{3}-1.

$$\frac{\sqrt{3}-1}{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}-\frac{\sqrt{3}}{2}+\sqrt{32}\sqrt{\frac{1}{8}}$$

Consider \left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\sqrt{3}-1}{\left(\sqrt{3}\right)^{2}-1^{2}}-\frac{\sqrt{3}}{2}+\sqrt{32}\sqrt{\frac{1}{8}}$$

Square \sqrt{3}. Square 1.

$$\frac{\sqrt{3}-1}{3-1}-\frac{\sqrt{3}}{2}+\sqrt{32}\sqrt{\frac{1}{8}}$$

Subtract 1 from 3 to get 2.

$$\frac{\sqrt{3}-1}{2}-\frac{\sqrt{3}}{2}+\sqrt{32}\sqrt{\frac{1}{8}}$$

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}.

$$\frac{\sqrt{3}-1}{2}-\frac{\sqrt{3}}{2}+4\sqrt{2}\sqrt{\frac{1}{8}}$$

Rewrite the square root of the division \sqrt{\frac{1}{8}} as the division of square roots \frac{\sqrt{1}}{\sqrt{8}}.

$$\frac{\sqrt{3}-1}{2}-\frac{\sqrt{3}}{2}+4\sqrt{2}\times \frac{\sqrt{1}}{\sqrt{8}}$$

Calculate the square root of 1 and get 1.

$$\frac{\sqrt{3}-1}{2}-\frac{\sqrt{3}}{2}+4\sqrt{2}\times \frac{1}{\sqrt{8}}$$

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}.

$$\frac{\sqrt{3}-1}{2}-\frac{\sqrt{3}}{2}+4\sqrt{2}\times \frac{1}{2\sqrt{2}}$$

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

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

The square of \sqrt{2} is 2.

$$\frac{\sqrt{3}-1}{2}-\frac{\sqrt{3}}{2}+4\sqrt{2}\times \frac{\sqrt{2}}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{\sqrt{3}-1}{2}-\frac{\sqrt{3}}{2}+4\sqrt{2}\times \frac{\sqrt{2}}{4}$$

Cancel out 4 and 4.

$$\frac{\sqrt{3}-1}{2}-\frac{\sqrt{3}}{2}+\sqrt{2}\sqrt{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{\sqrt{3}-1}{2}-\frac{\sqrt{3}}{2}+2$$

Since \frac{\sqrt{3}-1}{2} and \frac{\sqrt{3}}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\sqrt{3}-1-\sqrt{3}}{2}+2$$

Do the calculations in \sqrt{3}-1-\sqrt{3}.

$$\frac{-1}{2}+2$$

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

$$\frac{-1}{2}+\frac{2\times 2}{2}$$

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

$$\frac{-1+2\times 2}{2}$$

Do the multiplications in -1+2\times 2.

$$\frac{-1+4}{2}$$

Do the calculations in -1+4.

$$\frac{3}{2}$$

Factor

$$\frac{3}{2} = 1\frac{1}{2} = 1.5$$