Solve frac{7sqrt{3}}{sqrt{5}+2sqrt{3}}

$$\frac { 7 \sqrt { 3 } } { \sqrt { 5 } + 2 \sqrt { 3 } }$$

Evaluate

$$6-\sqrt{15}\approx 2.127016654$$

Solution Steps

Rationalize the denominator of \frac{7\sqrt{3}}{\sqrt{5}+2\sqrt{3}} by multiplying numerator and denominator by \sqrt{5}-2\sqrt{3}.

$$\frac{7\sqrt{3}\left(\sqrt{5}-2\sqrt{3}\right)}{\left(\sqrt{5}+2\sqrt{3}\right)\left(\sqrt{5}-2\sqrt{3}\right)}$$

Consider \left(\sqrt{5}+2\sqrt{3}\right)\left(\sqrt{5}-2\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{7\sqrt{3}\left(\sqrt{5}-2\sqrt{3}\right)}{\left(\sqrt{5}\right)^{2}-\left(2\sqrt{3}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{7\sqrt{3}\left(\sqrt{5}-2\sqrt{3}\right)}{5-\left(2\sqrt{3}\right)^{2}}$$

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

$$\frac{7\sqrt{3}\left(\sqrt{5}-2\sqrt{3}\right)}{5-2^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{7\sqrt{3}\left(\sqrt{5}-2\sqrt{3}\right)}{5-4\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{7\sqrt{3}\left(\sqrt{5}-2\sqrt{3}\right)}{5-4\times 3}$$

Multiply 4 and 3 to get 12.

$$\frac{7\sqrt{3}\left(\sqrt{5}-2\sqrt{3}\right)}{5-12}$$

Subtract 12 from 5 to get -7.

$$\frac{7\sqrt{3}\left(\sqrt{5}-2\sqrt{3}\right)}{-7}$$

Cancel out -7 and -7.

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

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

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

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

$$-\sqrt{15}+2\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$-\sqrt{15}+2\times 3$$

Multiply 2 and 3 to get 6.

$$-\sqrt{15}+6$$