Solve -sqrt{343}-5sqrt{4/25}+sqrt{54}-1/4sqrt{112}

$$- \sqrt { 343 } - 5 \sqrt { \frac { 4 } { 25 } } + \sqrt { 54 } - \frac { 1 } { 4 } \sqrt { 112 }$$

Evaluate

$$3\sqrt{6}-8\sqrt{7}-2\approx -15.81754126$$

Solution Steps

Factor 343=7^{2}\times 7. Rewrite the square root of the product \sqrt{7^{2}\times 7} as the product of square roots \sqrt{7^{2}}\sqrt{7}. Take the square root of 7^{2}.

$$-7\sqrt{7}-5\sqrt{\frac{4}{25}}+\sqrt{54}-\frac{1}{4}\sqrt{112}$$

Rewrite the square root of the division \frac{4}{25} as the division of square roots \frac{\sqrt{4}}{\sqrt{25}}. Take the square root of both numerator and denominator.

$$-7\sqrt{7}-5\times \frac{2}{5}+\sqrt{54}-\frac{1}{4}\sqrt{112}$$

Multiply -5 times \frac{2}{5}.

$$-7\sqrt{7}-2+\sqrt{54}-\frac{1}{4}\sqrt{112}$$

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}.

$$-7\sqrt{7}-2+3\sqrt{6}-\frac{1}{4}\sqrt{112}$$

Factor 112=4^{2}\times 7. Rewrite the square root of the product \sqrt{4^{2}\times 7} as the product of square roots \sqrt{4^{2}}\sqrt{7}. Take the square root of 4^{2}.

$$-7\sqrt{7}-2+3\sqrt{6}-\frac{1}{4}\times 4\sqrt{7}$$

Cancel out 4 and 4.

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

Combine -7\sqrt{7} and -\sqrt{7} to get -8\sqrt{7}.

$$-8\sqrt{7}-2+3\sqrt{6}$$

Factor

$$3 \sqrt{6} - 8 \sqrt{7} - 2 = -15.81754126$$