Solve (sqrt{12}-sqrt{63}+sqrt{75})*sqrt{3}

$$( \sqrt { 12 } - \sqrt { 63 } + \sqrt { 75 } ) \cdot \sqrt { 3 }$$

Evaluate

$$21-3\sqrt{21}\approx 7.252272915$$

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}.

$$\left(2\sqrt{3}-\sqrt{63}+\sqrt{75}\right)\sqrt{3}$$

Factor 63=3^{2}\times 7. Rewrite the square root of the product \sqrt{3^{2}\times 7} as the product of square roots \sqrt{3^{2}}\sqrt{7}. Take the square root of 3^{2}.

$$\left(2\sqrt{3}-3\sqrt{7}+\sqrt{75}\right)\sqrt{3}$$

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

$$\left(2\sqrt{3}-3\sqrt{7}+5\sqrt{3}\right)\sqrt{3}$$

Combine 2\sqrt{3} and 5\sqrt{3} to get 7\sqrt{3}.

$$\left(7\sqrt{3}-3\sqrt{7}\right)\sqrt{3}$$

Use the distributive property to multiply 7\sqrt{3}-3\sqrt{7} by \sqrt{3}.

$$7\left(\sqrt{3}\right)^{2}-3\sqrt{7}\sqrt{3}$$

The square of \sqrt{3} is 3.

$$7\times 3-3\sqrt{7}\sqrt{3}$$

Multiply 7 and 3 to get 21.

$$21-3\sqrt{7}\sqrt{3}$$

To multiply \sqrt{7} and \sqrt{3}, multiply the numbers under the square root.

$$21-3\sqrt{21}$$