Solve frac{4sqrt{5}}{2sqrt{3}+sqrt{14}}

$$\frac { 4 \sqrt { 5 } } { 2 \sqrt { 3 } + \sqrt { 14 } }$$

Evaluate

$$2\sqrt{70}-4\sqrt{15}\approx 1.241267146$$

Solution Steps

Rationalize the denominator of \frac{4\sqrt{5}}{2\sqrt{3}+\sqrt{14}} by multiplying numerator and denominator by 2\sqrt{3}-\sqrt{14}.

$$\frac{4\sqrt{5}\left(2\sqrt{3}-\sqrt{14}\right)}{\left(2\sqrt{3}+\sqrt{14}\right)\left(2\sqrt{3}-\sqrt{14}\right)}$$

Consider \left(2\sqrt{3}+\sqrt{14}\right)\left(2\sqrt{3}-\sqrt{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}.

$$\frac{4\sqrt{5}\left(2\sqrt{3}-\sqrt{14}\right)}{\left(2\sqrt{3}\right)^{2}-\left(\sqrt{14}\right)^{2}}$$

Expand \left(2\sqrt{3}\right)^{2}.

$$\frac{4\sqrt{5}\left(2\sqrt{3}-\sqrt{14}\right)}{2^{2}\left(\sqrt{3}\right)^{2}-\left(\sqrt{14}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{4\sqrt{5}\left(2\sqrt{3}-\sqrt{14}\right)}{4\left(\sqrt{3}\right)^{2}-\left(\sqrt{14}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{4\sqrt{5}\left(2\sqrt{3}-\sqrt{14}\right)}{4\times 3-\left(\sqrt{14}\right)^{2}}$$

Multiply 4 and 3 to get 12.

$$\frac{4\sqrt{5}\left(2\sqrt{3}-\sqrt{14}\right)}{12-\left(\sqrt{14}\right)^{2}}$$

The square of \sqrt{14} is 14.

$$\frac{4\sqrt{5}\left(2\sqrt{3}-\sqrt{14}\right)}{12-14}$$

Subtract 14 from 12 to get -2.

$$\frac{4\sqrt{5}\left(2\sqrt{3}-\sqrt{14}\right)}{-2}$$

Divide 4\sqrt{5}\left(2\sqrt{3}-\sqrt{14}\right) by -2 to get -2\sqrt{5}\left(2\sqrt{3}-\sqrt{14}\right).

$$-2\sqrt{5}\left(2\sqrt{3}-\sqrt{14}\right)$$

Use the distributive property to multiply -2\sqrt{5} by 2\sqrt{3}-\sqrt{14}.

$$-4\sqrt{3}\sqrt{5}+2\sqrt{5}\sqrt{14}$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$-4\sqrt{15}+2\sqrt{5}\sqrt{14}$$

To multiply \sqrt{5} and \sqrt{14}, multiply the numbers under the square root.

$$-4\sqrt{15}+2\sqrt{70}$$