Solve frac{7sqrt{5}-sqrt{7}}{5sqrt{7}-sqrt{5}}

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

Evaluate

$$\frac{\sqrt{35}}{5}\approx 1.183215957$$

Solution Steps

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

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

Consider \left(5\sqrt{7}-\sqrt{5}\right)\left(5\sqrt{7}+\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(5\sqrt{7}\right)^{2}.

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

Calculate 5 to the power of 2 and get 25.

$$\frac{\left(7\sqrt{5}-\sqrt{7}\right)\left(5\sqrt{7}+\sqrt{5}\right)}{25\left(\sqrt{7}\right)^{2}-\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{\left(7\sqrt{5}-\sqrt{7}\right)\left(5\sqrt{7}+\sqrt{5}\right)}{25\times 7-\left(\sqrt{5}\right)^{2}}$$

Multiply 25 and 7 to get 175.

$$\frac{\left(7\sqrt{5}-\sqrt{7}\right)\left(5\sqrt{7}+\sqrt{5}\right)}{175-\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(7\sqrt{5}-\sqrt{7}\right)\left(5\sqrt{7}+\sqrt{5}\right)}{175-5}$$

Subtract 5 from 175 to get 170.

$$\frac{\left(7\sqrt{5}-\sqrt{7}\right)\left(5\sqrt{7}+\sqrt{5}\right)}{170}$$

Apply the distributive property by multiplying each term of 7\sqrt{5}-\sqrt{7} by each term of 5\sqrt{7}+\sqrt{5}.

$$\frac{35\sqrt{5}\sqrt{7}+7\left(\sqrt{5}\right)^{2}-5\left(\sqrt{7}\right)^{2}-\sqrt{7}\sqrt{5}}{170}$$

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

$$\frac{35\sqrt{35}+7\left(\sqrt{5}\right)^{2}-5\left(\sqrt{7}\right)^{2}-\sqrt{7}\sqrt{5}}{170}$$

The square of \sqrt{5} is 5.

$$\frac{35\sqrt{35}+7\times 5-5\left(\sqrt{7}\right)^{2}-\sqrt{7}\sqrt{5}}{170}$$

Multiply 7 and 5 to get 35.

$$\frac{35\sqrt{35}+35-5\left(\sqrt{7}\right)^{2}-\sqrt{7}\sqrt{5}}{170}$$

The square of \sqrt{7} is 7.

$$\frac{35\sqrt{35}+35-5\times 7-\sqrt{7}\sqrt{5}}{170}$$

Multiply -5 and 7 to get -35.

$$\frac{35\sqrt{35}+35-35-\sqrt{7}\sqrt{5}}{170}$$

Subtract 35 from 35 to get 0.

$$\frac{35\sqrt{35}-\sqrt{7}\sqrt{5}}{170}$$

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

$$\frac{35\sqrt{35}-\sqrt{35}}{170}$$

Combine 35\sqrt{35} and -\sqrt{35} to get 34\sqrt{35}.

$$\frac{34\sqrt{35}}{170}$$

Divide 34\sqrt{35} by 170 to get \frac{1}{5}\sqrt{35}.

$$\frac{1}{5}\sqrt{35}$$