Solve 1/sqrt{27+sqrt{12}+5sqrt{2}}-4/5sqrt{2}

$$\frac { 1 } { \sqrt { 27 } + \sqrt { 12 } + 5 \sqrt { 2 } } - \frac { 4 } { 5 \sqrt { 2 } }$$

Evaluate

$$\frac{\sqrt{3}-3\sqrt{2}}{5}\approx -0.502117976$$

Short Solution Steps

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$\frac{1}{3\sqrt{3}+\sqrt{12}+5\sqrt{2}}-\frac{4}{5\sqrt{2}}$$

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}.

$$\frac{1}{3\sqrt{3}+2\sqrt{3}+5\sqrt{2}}-\frac{4}{5\sqrt{2}}$$

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

$$\frac{1}{5\sqrt{3}+5\sqrt{2}}-\frac{4}{5\sqrt{2}}$$

Rationalize the denominator of \frac{1}{5\sqrt{3}+5\sqrt{2}} by multiplying numerator and denominator by 5\sqrt{3}-5\sqrt{2}.

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

Consider \left(5\sqrt{3}+5\sqrt{2}\right)\left(5\sqrt{3}-5\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(5\sqrt{3}\right)^{2}.

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

Calculate 5 to the power of 2 and get 25.

$$\frac{5\sqrt{3}-5\sqrt{2}}{25\left(\sqrt{3}\right)^{2}-\left(5\sqrt{2}\right)^{2}}-\frac{4}{5\sqrt{2}}$$

The square of \sqrt{3} is 3.

$$\frac{5\sqrt{3}-5\sqrt{2}}{25\times 3-\left(5\sqrt{2}\right)^{2}}-\frac{4}{5\sqrt{2}}$$

Multiply 25 and 3 to get 75.

$$\frac{5\sqrt{3}-5\sqrt{2}}{75-\left(5\sqrt{2}\right)^{2}}-\frac{4}{5\sqrt{2}}$$

Expand \left(5\sqrt{2}\right)^{2}.

$$\frac{5\sqrt{3}-5\sqrt{2}}{75-5^{2}\left(\sqrt{2}\right)^{2}}-\frac{4}{5\sqrt{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{5\sqrt{3}-5\sqrt{2}}{75-25\left(\sqrt{2}\right)^{2}}-\frac{4}{5\sqrt{2}}$$

The square of \sqrt{2} is 2.

$$\frac{5\sqrt{3}-5\sqrt{2}}{75-25\times 2}-\frac{4}{5\sqrt{2}}$$

Multiply 25 and 2 to get 50.

$$\frac{5\sqrt{3}-5\sqrt{2}}{75-50}-\frac{4}{5\sqrt{2}}$$

Subtract 50 from 75 to get 25.

$$\frac{5\sqrt{3}-5\sqrt{2}}{25}-\frac{4}{5\sqrt{2}}$$

Rationalize the denominator of \frac{4}{5\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{5\sqrt{3}-5\sqrt{2}}{25}-\frac{4\sqrt{2}}{5\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{5\sqrt{3}-5\sqrt{2}}{25}-\frac{4\sqrt{2}}{5\times 2}$$

Cancel out 2 in both numerator and denominator.

$$\frac{5\sqrt{3}-5\sqrt{2}}{25}-\frac{2\sqrt{2}}{5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 25 and 5 is 25. Multiply \frac{2\sqrt{2}}{5} times \frac{5}{5}.

$$\frac{5\sqrt{3}-5\sqrt{2}}{25}-\frac{5\times 2\sqrt{2}}{25}$$

Since \frac{5\sqrt{3}-5\sqrt{2}}{25} and \frac{5\times 2\sqrt{2}}{25} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5\sqrt{3}-5\sqrt{2}-5\times 2\sqrt{2}}{25}$$

Do the multiplications in 5\sqrt{3}-5\sqrt{2}-5\times 2\sqrt{2}.

$$\frac{5\sqrt{3}-5\sqrt{2}-10\sqrt{2}}{25}$$

Do the calculations in 5\sqrt{3}-5\sqrt{2}-10\sqrt{2}.

$$\frac{5\sqrt{3}-15\sqrt{2}}{25}$$

Factor

$$\frac{\sqrt{3} - 3 \sqrt{2}}{5} = -0.5021179759100817$$