Solve (1/2sqrt{28}-3/2sqrt{84})*sqrt{14}

$$( \frac { 1 } { 2 } \sqrt { 28 } - \frac { 3 } { 2 } \sqrt { 84 } ) \times \sqrt { 14 }$$

Evaluate

$$7\left(\sqrt{2}-3\sqrt{6}\right)\approx -41.539789662$$

Solution Steps

Factor 28=2^{2}\times 7. Rewrite the square root of the product \sqrt{2^{2}\times 7} as the product of square roots \sqrt{2^{2}}\sqrt{7}. Take the square root of 2^{2}.

$$\left(\frac{1}{2}\times 2\sqrt{7}-\frac{3}{2}\sqrt{84}\right)\sqrt{14}$$

Cancel out 2 and 2.

$$\left(\sqrt{7}-\frac{3}{2}\sqrt{84}\right)\sqrt{14}$$

Factor 84=2^{2}\times 21. Rewrite the square root of the product \sqrt{2^{2}\times 21} as the product of square roots \sqrt{2^{2}}\sqrt{21}. Take the square root of 2^{2}.

$$\left(\sqrt{7}-\frac{3}{2}\times 2\sqrt{21}\right)\sqrt{14}$$

Cancel out 2 and 2.

$$\left(\sqrt{7}-3\sqrt{21}\right)\sqrt{14}$$

Use the distributive property to multiply \sqrt{7}-3\sqrt{21} by \sqrt{14}.

$$\sqrt{7}\sqrt{14}-3\sqrt{21}\sqrt{14}$$

Factor 14=7\times 2. Rewrite the square root of the product \sqrt{7\times 2} as the product of square roots \sqrt{7}\sqrt{2}.

$$\sqrt{7}\sqrt{7}\sqrt{2}-3\sqrt{21}\sqrt{14}$$

Multiply \sqrt{7} and \sqrt{7} to get 7.

$$7\sqrt{2}-3\sqrt{21}\sqrt{14}$$

To multiply \sqrt{21} and \sqrt{14}, multiply the numbers under the square root.

$$7\sqrt{2}-3\sqrt{294}$$

Factor 294=7^{2}\times 6. Rewrite the square root of the product \sqrt{7^{2}\times 6} as the product of square roots \sqrt{7^{2}}\sqrt{6}. Take the square root of 7^{2}.

$$7\sqrt{2}-3\times 7\sqrt{6}$$

Multiply -3 and 7 to get -21.

$$7\sqrt{2}-21\sqrt{6}$$

Factor

$$7 {(\sqrt{2} - 3 \sqrt{6})} = -41.539789662$$