Solve frac{sqrt{63}+sqrt{28}}{sqrt{150}}*frac{sqrt{384}-sqrt{294}}{sqrt{112}}

$$\frac { \sqrt { 63 } + \sqrt { 28 } } { \sqrt { 150 } } \times \frac { \sqrt { 384 } - \sqrt { 294 } } { \sqrt { 112 } }$$

Evaluate

$$\frac{1}{4}=0.25$$

Solution Steps

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}.

$$\frac{3\sqrt{7}+\sqrt{28}}{\sqrt{150}}\times \frac{\sqrt{384}-\sqrt{294}}{\sqrt{112}}$$

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}.

$$\frac{3\sqrt{7}+2\sqrt{7}}{\sqrt{150}}\times \frac{\sqrt{384}-\sqrt{294}}{\sqrt{112}}$$

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

$$\frac{5\sqrt{7}}{\sqrt{150}}\times \frac{\sqrt{384}-\sqrt{294}}{\sqrt{112}}$$

Factor 150=5^{2}\times 6. Rewrite the square root of the product \sqrt{5^{2}\times 6} as the product of square roots \sqrt{5^{2}}\sqrt{6}. Take the square root of 5^{2}.

$$\frac{5\sqrt{7}}{5\sqrt{6}}\times \frac{\sqrt{384}-\sqrt{294}}{\sqrt{112}}$$

Cancel out 5 in both numerator and denominator.

$$\frac{\sqrt{7}}{\sqrt{6}}\times \frac{\sqrt{384}-\sqrt{294}}{\sqrt{112}}$$

Rationalize the denominator of \frac{\sqrt{7}}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{\sqrt{7}\sqrt{6}}{\left(\sqrt{6}\right)^{2}}\times \frac{\sqrt{384}-\sqrt{294}}{\sqrt{112}}$$

The square of \sqrt{6} is 6.

$$\frac{\sqrt{7}\sqrt{6}}{6}\times \frac{\sqrt{384}-\sqrt{294}}{\sqrt{112}}$$

To multiply \sqrt{7} and \sqrt{6}, multiply the numbers under the square root.

$$\frac{\sqrt{42}}{6}\times \frac{\sqrt{384}-\sqrt{294}}{\sqrt{112}}$$

Factor 384=8^{2}\times 6. Rewrite the square root of the product \sqrt{8^{2}\times 6} as the product of square roots \sqrt{8^{2}}\sqrt{6}. Take the square root of 8^{2}.

$$\frac{\sqrt{42}}{6}\times \frac{8\sqrt{6}-\sqrt{294}}{\sqrt{112}}$$

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}.

$$\frac{\sqrt{42}}{6}\times \frac{8\sqrt{6}-7\sqrt{6}}{\sqrt{112}}$$

Combine 8\sqrt{6} and -7\sqrt{6} to get \sqrt{6}.

$$\frac{\sqrt{42}}{6}\times \frac{\sqrt{6}}{\sqrt{112}}$$

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}.

$$\frac{\sqrt{42}}{6}\times \frac{\sqrt{6}}{4\sqrt{7}}$$

Rationalize the denominator of \frac{\sqrt{6}}{4\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\frac{\sqrt{42}}{6}\times \frac{\sqrt{6}\sqrt{7}}{4\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{\sqrt{42}}{6}\times \frac{\sqrt{6}\sqrt{7}}{4\times 7}$$

To multiply \sqrt{6} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{\sqrt{42}}{6}\times \frac{\sqrt{42}}{4\times 7}$$

Multiply 4 and 7 to get 28.

$$\frac{\sqrt{42}}{6}\times \frac{\sqrt{42}}{28}$$

Multiply \frac{\sqrt{42}}{6} times \frac{\sqrt{42}}{28} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{42}\sqrt{42}}{6\times 28}$$

Multiply \sqrt{42} and \sqrt{42} to get 42.

$$\frac{42}{6\times 28}$$

Multiply 6 and 28 to get 168.

$$\frac{42}{168}$$

Reduce the fraction \frac{42}{168} to lowest terms by extracting and canceling out 42.

$$\frac{1}{4}$$

Factor

$$\frac{1}{2 ^ {2}} = 0.25$$