Solve (7sqrt{54}-3sqrt{24}+6)divsqrt{3}

$$(7 \sqrt{ 54 } -3 \sqrt{ 24 } +6) \div \sqrt{ 3 }$$

Evaluate

$$\sqrt{2}\left(\sqrt{6}+15\right)\approx 24.677305051$$

Solution Steps

Factor 54=3^{2}\times 6. Rewrite the square root of the product \sqrt{3^{2}\times 6} as the product of square roots \sqrt{3^{2}}\sqrt{6}. Take the square root of 3^{2}.

$$\frac{7\times 3\sqrt{6}-3\sqrt{24}+6}{\sqrt{3}}$$

Multiply 7 and 3 to get 21.

$$\frac{21\sqrt{6}-3\sqrt{24}+6}{\sqrt{3}}$$

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

$$\frac{21\sqrt{6}-3\times 2\sqrt{6}+6}{\sqrt{3}}$$

Multiply -3 and 2 to get -6.

$$\frac{21\sqrt{6}-6\sqrt{6}+6}{\sqrt{3}}$$

Combine 21\sqrt{6} and -6\sqrt{6} to get 15\sqrt{6}.

$$\frac{15\sqrt{6}+6}{\sqrt{3}}$$

Rationalize the denominator of \frac{15\sqrt{6}+6}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

$$\frac{\left(15\sqrt{6}+6\right)\sqrt{3}}{3}$$

Use the distributive property to multiply 15\sqrt{6}+6 by \sqrt{3}.

$$\frac{15\sqrt{6}\sqrt{3}+6\sqrt{3}}{3}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$\frac{15\sqrt{3}\sqrt{2}\sqrt{3}+6\sqrt{3}}{3}$$

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

$$\frac{15\times 3\sqrt{2}+6\sqrt{3}}{3}$$

Multiply 15 and 3 to get 45.

$$\frac{45\sqrt{2}+6\sqrt{3}}{3}$$

Divide each term of 45\sqrt{2}+6\sqrt{3} by 3 to get 15\sqrt{2}+2\sqrt{3}.

$$15\sqrt{2}+2\sqrt{3}$$