Solve 5/6+4sqrt{7}

$$\frac{ 5 }{ 6+4 \sqrt{ 7 } }$$

Evaluate

$$\frac{5\sqrt{7}}{19}-\frac{15}{38}\approx 0.301513503$$

Solution Steps

Rationalize the denominator of \frac{5}{6+4\sqrt{7}} by multiplying numerator and denominator by 6-4\sqrt{7}.

$$\frac{5\left(6-4\sqrt{7}\right)}{\left(6+4\sqrt{7}\right)\left(6-4\sqrt{7}\right)}$$

Consider \left(6+4\sqrt{7}\right)\left(6-4\sqrt{7}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{5\left(6-4\sqrt{7}\right)}{6^{2}-\left(4\sqrt{7}\right)^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{5\left(6-4\sqrt{7}\right)}{36-\left(4\sqrt{7}\right)^{2}}$$

Expand \left(4\sqrt{7}\right)^{2}.

$$\frac{5\left(6-4\sqrt{7}\right)}{36-4^{2}\left(\sqrt{7}\right)^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{5\left(6-4\sqrt{7}\right)}{36-16\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{5\left(6-4\sqrt{7}\right)}{36-16\times 7}$$

Multiply 16 and 7 to get 112.

$$\frac{5\left(6-4\sqrt{7}\right)}{36-112}$$

Subtract 112 from 36 to get -76.

$$\frac{5\left(6-4\sqrt{7}\right)}{-76}$$

Use the distributive property to multiply 5 by 6-4\sqrt{7}.

$$\frac{30-20\sqrt{7}}{-76}$$