Solve frac{10sqrt{6}+2sqrt{3}}{sqrt{6}-sqrt{3}}

$$\frac{ 10 \sqrt{ 6 } +2 \sqrt{ 3 } }{ \sqrt{ 6 } - \sqrt{ 3 } }$$

Evaluate

$$12\sqrt{2}+22\approx 38.970562748$$

Solution Steps

Rationalize the denominator of \frac{10\sqrt{6}+2\sqrt{3}}{\sqrt{6}-\sqrt{3}} by multiplying numerator and denominator by \sqrt{6}+\sqrt{3}.

$$\frac{\left(10\sqrt{6}+2\sqrt{3}\right)\left(\sqrt{6}+\sqrt{3}\right)}{\left(\sqrt{6}-\sqrt{3}\right)\left(\sqrt{6}+\sqrt{3}\right)}$$

Consider \left(\sqrt{6}-\sqrt{3}\right)\left(\sqrt{6}+\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}.

$$\frac{\left(10\sqrt{6}+2\sqrt{3}\right)\left(\sqrt{6}+\sqrt{3}\right)}{\left(\sqrt{6}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

Square \sqrt{6}. Square \sqrt{3}.

$$\frac{\left(10\sqrt{6}+2\sqrt{3}\right)\left(\sqrt{6}+\sqrt{3}\right)}{6-3}$$

Subtract 3 from 6 to get 3.

$$\frac{\left(10\sqrt{6}+2\sqrt{3}\right)\left(\sqrt{6}+\sqrt{3}\right)}{3}$$

Apply the distributive property by multiplying each term of 10\sqrt{6}+2\sqrt{3} by each term of \sqrt{6}+\sqrt{3}.

$$\frac{10\left(\sqrt{6}\right)^{2}+10\sqrt{6}\sqrt{3}+2\sqrt{3}\sqrt{6}+2\left(\sqrt{3}\right)^{2}}{3}$$

The square of \sqrt{6} is 6.

$$\frac{10\times 6+10\sqrt{6}\sqrt{3}+2\sqrt{3}\sqrt{6}+2\left(\sqrt{3}\right)^{2}}{3}$$

Multiply 10 and 6 to get 60.

$$\frac{60+10\sqrt{6}\sqrt{3}+2\sqrt{3}\sqrt{6}+2\left(\sqrt{3}\right)^{2}}{3}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$\frac{60+10\sqrt{3}\sqrt{2}\sqrt{3}+2\sqrt{3}\sqrt{6}+2\left(\sqrt{3}\right)^{2}}{3}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{60+10\times 3\sqrt{2}+2\sqrt{3}\sqrt{6}+2\left(\sqrt{3}\right)^{2}}{3}$$

Multiply 10 and 3 to get 30.

$$\frac{60+30\sqrt{2}+2\sqrt{3}\sqrt{6}+2\left(\sqrt{3}\right)^{2}}{3}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$\frac{60+30\sqrt{2}+2\sqrt{3}\sqrt{3}\sqrt{2}+2\left(\sqrt{3}\right)^{2}}{3}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{60+30\sqrt{2}+2\times 3\sqrt{2}+2\left(\sqrt{3}\right)^{2}}{3}$$

Multiply 2 and 3 to get 6.

$$\frac{60+30\sqrt{2}+6\sqrt{2}+2\left(\sqrt{3}\right)^{2}}{3}$$

Combine 30\sqrt{2} and 6\sqrt{2} to get 36\sqrt{2}.

$$\frac{60+36\sqrt{2}+2\left(\sqrt{3}\right)^{2}}{3}$$

The square of \sqrt{3} is 3.

$$\frac{60+36\sqrt{2}+2\times 3}{3}$$

Multiply 2 and 3 to get 6.

$$\frac{60+36\sqrt{2}+6}{3}$$

Add 60 and 6 to get 66.

$$\frac{66+36\sqrt{2}}{3}$$

Factor

$$2 {(6 \sqrt{2} + 11)} = 38.970562748$$