Solve 7/7+7sqrt{5}

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

Evaluate

$$\frac{\sqrt{5}-1}{4}\approx 0.309016994$$

Solution Steps

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

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

Consider \left(7+7\sqrt{5}\right)\left(7-7\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}.

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

Calculate 7 to the power of 2 and get 49.

$$\frac{7\left(7-7\sqrt{5}\right)}{49-\left(7\sqrt{5}\right)^{2}}$$

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

$$\frac{7\left(7-7\sqrt{5}\right)}{49-7^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate 7 to the power of 2 and get 49.

$$\frac{7\left(7-7\sqrt{5}\right)}{49-49\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{7\left(7-7\sqrt{5}\right)}{49-49\times 5}$$

Multiply 49 and 5 to get 245.

$$\frac{7\left(7-7\sqrt{5}\right)}{49-245}$$

Subtract 245 from 49 to get -196.

$$\frac{7\left(7-7\sqrt{5}\right)}{-196}$$

Divide 7\left(7-7\sqrt{5}\right) by -196 to get -\frac{1}{28}\left(7-7\sqrt{5}\right).

$$-\frac{1}{28}\left(7-7\sqrt{5}\right)$$

Use the distributive property to multiply -\frac{1}{28} by 7-7\sqrt{5}.

$$-\frac{1}{28}\times 7-\frac{1}{28}\left(-7\right)\sqrt{5}$$

Express -\frac{1}{28}\times 7 as a single fraction.

$$\frac{-7}{28}-\frac{1}{28}\left(-7\right)\sqrt{5}$$

Reduce the fraction \frac{-7}{28} to lowest terms by extracting and canceling out 7.

$$-\frac{1}{4}-\frac{1}{28}\left(-7\right)\sqrt{5}$$

Express -\frac{1}{28}\left(-7\right) as a single fraction.

$$-\frac{1}{4}+\frac{-\left(-7\right)}{28}\sqrt{5}$$

Multiply -1 and -7 to get 7.

$$-\frac{1}{4}+\frac{7}{28}\sqrt{5}$$

Reduce the fraction \frac{7}{28} to lowest terms by extracting and canceling out 7.

$$-\frac{1}{4}+\frac{1}{4}\sqrt{5}$$