Solve frac{2sqrt{7}+1}{3}*frac{15sqrt{7}}{sqrt{7}+14}

$$\frac{ 2 \sqrt{ 7 } +1 }{ 3 } \times \frac{ 15 \sqrt{ 7 } }{ \sqrt{ 7 } +14 }$$

Evaluate

$$5$$

Solution Steps

Rationalize the denominator of \frac{15\sqrt{7}}{\sqrt{7}+14} by multiplying numerator and denominator by \sqrt{7}-14.

$$\frac{2\sqrt{7}+1}{3}\times \frac{15\sqrt{7}\left(\sqrt{7}-14\right)}{\left(\sqrt{7}+14\right)\left(\sqrt{7}-14\right)}$$

Consider \left(\sqrt{7}+14\right)\left(\sqrt{7}-14\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\sqrt{7}+1}{3}\times \frac{15\sqrt{7}\left(\sqrt{7}-14\right)}{\left(\sqrt{7}\right)^{2}-14^{2}}$$

Square \sqrt{7}. Square 14.

$$\frac{2\sqrt{7}+1}{3}\times \frac{15\sqrt{7}\left(\sqrt{7}-14\right)}{7-196}$$

Subtract 196 from 7 to get -189.

$$\frac{2\sqrt{7}+1}{3}\times \frac{15\sqrt{7}\left(\sqrt{7}-14\right)}{-189}$$

Multiply \frac{2\sqrt{7}+1}{3} times \frac{15\sqrt{7}\left(\sqrt{7}-14\right)}{-189} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(2\sqrt{7}+1\right)\times 15\sqrt{7}\left(\sqrt{7}-14\right)}{3\left(-189\right)}$$

Cancel out 3 in both numerator and denominator.

$$\frac{5\sqrt{7}\left(\sqrt{7}-14\right)\left(2\sqrt{7}+1\right)}{-189}$$

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

$$\frac{\left(5\left(\sqrt{7}\right)^{2}-70\sqrt{7}\right)\left(2\sqrt{7}+1\right)}{-189}$$

The square of \sqrt{7} is 7.

$$\frac{\left(5\times 7-70\sqrt{7}\right)\left(2\sqrt{7}+1\right)}{-189}$$

Multiply 5 and 7 to get 35.

$$\frac{\left(35-70\sqrt{7}\right)\left(2\sqrt{7}+1\right)}{-189}$$

Apply the distributive property by multiplying each term of 35-70\sqrt{7} by each term of 2\sqrt{7}+1.

$$\frac{70\sqrt{7}+35-140\left(\sqrt{7}\right)^{2}-70\sqrt{7}}{-189}$$

The square of \sqrt{7} is 7.

$$\frac{70\sqrt{7}+35-140\times 7-70\sqrt{7}}{-189}$$

Multiply -140 and 7 to get -980.

$$\frac{70\sqrt{7}+35-980-70\sqrt{7}}{-189}$$

Subtract 980 from 35 to get -945.

$$\frac{70\sqrt{7}-945-70\sqrt{7}}{-189}$$

Combine 70\sqrt{7} and -70\sqrt{7} to get 0.

$$\frac{-945}{-189}$$

Divide -945 by -189 to get 5.

$$5$$

Factor

$$5$$