Solve frac{sqrt{5}*sqrt{15}}{sqrt{3}}quadtext{(2)}quad(sqrt{3}+1)^2-6sqrt{1/3}

$$\frac { \sqrt { 5 } \times \sqrt { 15 } } { \sqrt { 3 } } \quad \text { (2) } \quad ( \sqrt { 3 } + 1 ) ^ { 2 } - 6 \sqrt { \frac { 1 } { 3 } }$$

Evaluate

$$18\sqrt{3}+40\approx 71.176914536$$

Solution Steps

Factor 15=5\times 3. Rewrite the square root of the product \sqrt{5\times 3} as the product of square roots \sqrt{5}\sqrt{3}.

$$\frac{\sqrt{5}\sqrt{5}\sqrt{3}}{\sqrt{3}}\times 2\left(\sqrt{3}+1\right)^{2}-6\sqrt{\frac{1}{3}}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{5\sqrt{3}}{\sqrt{3}}\times 2\left(\sqrt{3}+1\right)^{2}-6\sqrt{\frac{1}{3}}$$

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

$$\frac{5\sqrt{3}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\times 2\left(\sqrt{3}+1\right)^{2}-6\sqrt{\frac{1}{3}}$$

The square of \sqrt{3} is 3.

$$\frac{5\sqrt{3}\sqrt{3}}{3}\times 2\left(\sqrt{3}+1\right)^{2}-6\sqrt{\frac{1}{3}}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{5\times 3}{3}\times 2\left(\sqrt{3}+1\right)^{2}-6\sqrt{\frac{1}{3}}$$

Multiply 5 and 3 to get 15.

$$\frac{15}{3}\times 2\left(\sqrt{3}+1\right)^{2}-6\sqrt{\frac{1}{3}}$$

Divide 15 by 3 to get 5.

$$5\times 2\left(\sqrt{3}+1\right)^{2}-6\sqrt{\frac{1}{3}}$$

Multiply 5 and 2 to get 10.

$$10\left(\sqrt{3}+1\right)^{2}-6\sqrt{\frac{1}{3}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{3}+1\right)^{2}.

$$10\left(\left(\sqrt{3}\right)^{2}+2\sqrt{3}+1\right)-6\sqrt{\frac{1}{3}}$$

The square of \sqrt{3} is 3.

$$10\left(3+2\sqrt{3}+1\right)-6\sqrt{\frac{1}{3}}$$

Add 3 and 1 to get 4.

$$10\left(4+2\sqrt{3}\right)-6\sqrt{\frac{1}{3}}$$

Use the distributive property to multiply 10 by 4+2\sqrt{3}.

$$40+20\sqrt{3}-6\sqrt{\frac{1}{3}}$$

Rewrite the square root of the division \sqrt{\frac{1}{3}} as the division of square roots \frac{\sqrt{1}}{\sqrt{3}}.

$$40+20\sqrt{3}-6\times \frac{\sqrt{1}}{\sqrt{3}}$$

Calculate the square root of 1 and get 1.

$$40+20\sqrt{3}-6\times \frac{1}{\sqrt{3}}$$

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

$$40+20\sqrt{3}-6\times \frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$40+20\sqrt{3}-6\times \frac{\sqrt{3}}{3}$$

Cancel out 3, the greatest common factor in 6 and 3.

$$40+20\sqrt{3}-2\sqrt{3}$$

Combine 20\sqrt{3} and -2\sqrt{3} to get 18\sqrt{3}.

$$40+18\sqrt{3}$$

Factor

$$2 {(9 \sqrt{3} + 20)} = 71.176914536$$