Solve frac{(sqrt{5}-sqrt{7})}{sqrt{7}}-frac{(sqrt{5}-sqrt{7})}{sqrt{28}}=

$$\frac { ( \sqrt { 5 } - \sqrt { 7 } ) } { \sqrt { 7 } } - \frac { ( \sqrt { 5 } - \sqrt { 7 } ) } { \sqrt { 28 } } =$$

Evaluate

$$\frac{\sqrt{35}}{14}-\frac{1}{2}\approx -0.077422873$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{5}-\sqrt{7}}{\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\frac{\left(\sqrt{5}-\sqrt{7}\right)\sqrt{7}}{\left(\sqrt{7}\right)^{2}}-\frac{\sqrt{5}-\sqrt{7}}{\sqrt{28}}$$

The square of \sqrt{7} is 7.

$$\frac{\left(\sqrt{5}-\sqrt{7}\right)\sqrt{7}}{7}-\frac{\sqrt{5}-\sqrt{7}}{\sqrt{28}}$$

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}.

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

Rationalize the denominator of \frac{\sqrt{5}-\sqrt{7}}{2\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\frac{\left(\sqrt{5}-\sqrt{7}\right)\sqrt{7}}{7}-\frac{\left(\sqrt{5}-\sqrt{7}\right)\sqrt{7}}{2\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{\left(\sqrt{5}-\sqrt{7}\right)\sqrt{7}}{7}-\frac{\left(\sqrt{5}-\sqrt{7}\right)\sqrt{7}}{2\times 7}$$

Multiply 2 and 7 to get 14.

$$\frac{\left(\sqrt{5}-\sqrt{7}\right)\sqrt{7}}{7}-\frac{\left(\sqrt{5}-\sqrt{7}\right)\sqrt{7}}{14}$$

Combine \frac{\left(\sqrt{5}-\sqrt{7}\right)\sqrt{7}}{7} and -\frac{\left(\sqrt{5}-\sqrt{7}\right)\sqrt{7}}{14} to get \frac{1}{14}\left(\sqrt{5}-\sqrt{7}\right)\sqrt{7}.

$$\frac{1}{14}\left(\sqrt{5}-\sqrt{7}\right)\sqrt{7}$$

Use the distributive property to multiply \frac{1}{14} by \sqrt{5}-\sqrt{7}.

$$\left(\frac{1}{14}\sqrt{5}+\frac{1}{14}\left(-1\right)\sqrt{7}\right)\sqrt{7}$$

Multiply \frac{1}{14} and -1 to get -\frac{1}{14}.

$$\left(\frac{1}{14}\sqrt{5}-\frac{1}{14}\sqrt{7}\right)\sqrt{7}$$

Use the distributive property to multiply \frac{1}{14}\sqrt{5}-\frac{1}{14}\sqrt{7} by \sqrt{7}.

$$\frac{1}{14}\sqrt{5}\sqrt{7}-\frac{1}{14}\sqrt{7}\sqrt{7}$$

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

$$\frac{1}{14}\sqrt{5}\sqrt{7}-\frac{1}{14}\times 7$$

To multiply \sqrt{5} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{1}{14}\sqrt{35}-\frac{1}{14}\times 7$$

Express -\frac{1}{14}\times 7 as a single fraction.

$$\frac{1}{14}\sqrt{35}+\frac{-7}{14}$$

Reduce the fraction \frac{-7}{14} to lowest terms by extracting and canceling out 7.

$$\frac{1}{14}\sqrt{35}-\frac{1}{2}$$

Factor

$$\frac{\sqrt{35} - 7}{14} = -0.0774228726357417$$