Solve (7sqrt{3}-2sqrt{12}+3sqrt{18})divsqrt{1/3}

$$(7 \sqrt{ 3 } -2 \sqrt{ 12 } +3 \sqrt{ 18 } ) \div \sqrt{ \frac{ 1 }{ 3 } }$$

Evaluate

$$9\left(\sqrt{6}+1\right)\approx 31.045407685$$

Solution Steps

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}.

$$\frac{7\sqrt{3}-2\times 2\sqrt{3}+3\sqrt{18}}{\sqrt{\frac{1}{3}}}$$

Multiply -2 and 2 to get -4.

$$\frac{7\sqrt{3}-4\sqrt{3}+3\sqrt{18}}{\sqrt{\frac{1}{3}}}$$

Combine 7\sqrt{3} and -4\sqrt{3} to get 3\sqrt{3}.

$$\frac{3\sqrt{3}+3\sqrt{18}}{\sqrt{\frac{1}{3}}}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

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

Multiply 3 and 3 to get 9.

$$\frac{3\sqrt{3}+9\sqrt{2}}{\sqrt{\frac{1}{3}}}$$

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

$$\frac{3\sqrt{3}+9\sqrt{2}}{\frac{\sqrt{1}}{\sqrt{3}}}$$

Calculate the square root of 1 and get 1.

$$\frac{3\sqrt{3}+9\sqrt{2}}{\frac{1}{\sqrt{3}}}$$

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

$$\frac{3\sqrt{3}+9\sqrt{2}}{\frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}$$

The square of \sqrt{3} is 3.

$$\frac{3\sqrt{3}+9\sqrt{2}}{\frac{\sqrt{3}}{3}}$$

Divide 3\sqrt{3}+9\sqrt{2} by \frac{\sqrt{3}}{3} by multiplying 3\sqrt{3}+9\sqrt{2} by the reciprocal of \frac{\sqrt{3}}{3}.

$$\frac{\left(3\sqrt{3}+9\sqrt{2}\right)\times 3}{\sqrt{3}}$$

Rationalize the denominator of \frac{\left(3\sqrt{3}+9\sqrt{2}\right)\times 3}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\left(3\sqrt{3}+9\sqrt{2}\right)\times 3\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(3\sqrt{3}+9\sqrt{2}\right)\times 3\sqrt{3}}{3}$$

Use the distributive property to multiply 3\sqrt{3}+9\sqrt{2} by 3.

$$\frac{\left(9\sqrt{3}+27\sqrt{2}\right)\sqrt{3}}{3}$$

Use the distributive property to multiply 9\sqrt{3}+27\sqrt{2} by \sqrt{3}.

$$\frac{9\left(\sqrt{3}\right)^{2}+27\sqrt{2}\sqrt{3}}{3}$$

The square of \sqrt{3} is 3.

$$\frac{9\times 3+27\sqrt{2}\sqrt{3}}{3}$$

Multiply 9 and 3 to get 27.

$$\frac{27+27\sqrt{2}\sqrt{3}}{3}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{27+27\sqrt{6}}{3}$$