Solve frac{11-sqrt{11}}{sqrt{33}-sqrt{3}}*frac{sqrt{3}}{sqrt{11}}=?

$$\frac { 11 - \sqrt { 11 } } { \sqrt { 33 } - \sqrt { 3 } } \cdot \frac { \sqrt { 3 } } { \sqrt { 11 } } = ?$$

Evaluate

$$1$$

Solution Steps

Rationalize the denominator of \frac{11-\sqrt{11}}{\sqrt{33}-\sqrt{3}} by multiplying numerator and denominator by \sqrt{33}+\sqrt{3}.

$$\frac{\left(11-\sqrt{11}\right)\left(\sqrt{33}+\sqrt{3}\right)}{\left(\sqrt{33}-\sqrt{3}\right)\left(\sqrt{33}+\sqrt{3}\right)}\times \frac{\sqrt{3}}{\sqrt{11}}$$

Consider \left(\sqrt{33}-\sqrt{3}\right)\left(\sqrt{33}+\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}.

$$\frac{\left(11-\sqrt{11}\right)\left(\sqrt{33}+\sqrt{3}\right)}{\left(\sqrt{33}\right)^{2}-\left(\sqrt{3}\right)^{2}}\times \frac{\sqrt{3}}{\sqrt{11}}$$

Square \sqrt{33}. Square \sqrt{3}.

$$\frac{\left(11-\sqrt{11}\right)\left(\sqrt{33}+\sqrt{3}\right)}{33-3}\times \frac{\sqrt{3}}{\sqrt{11}}$$

Subtract 3 from 33 to get 30.

$$\frac{\left(11-\sqrt{11}\right)\left(\sqrt{33}+\sqrt{3}\right)}{30}\times \frac{\sqrt{3}}{\sqrt{11}}$$

Rationalize the denominator of \frac{\sqrt{3}}{\sqrt{11}} by multiplying numerator and denominator by \sqrt{11}.

$$\frac{\left(11-\sqrt{11}\right)\left(\sqrt{33}+\sqrt{3}\right)}{30}\times \frac{\sqrt{3}\sqrt{11}}{\left(\sqrt{11}\right)^{2}}$$

The square of \sqrt{11} is 11.

$$\frac{\left(11-\sqrt{11}\right)\left(\sqrt{33}+\sqrt{3}\right)}{30}\times \frac{\sqrt{3}\sqrt{11}}{11}$$

To multiply \sqrt{3} and \sqrt{11}, multiply the numbers under the square root.

$$\frac{\left(11-\sqrt{11}\right)\left(\sqrt{33}+\sqrt{3}\right)}{30}\times \frac{\sqrt{33}}{11}$$

Multiply \frac{\left(11-\sqrt{11}\right)\left(\sqrt{33}+\sqrt{3}\right)}{30} times \frac{\sqrt{33}}{11} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(11-\sqrt{11}\right)\left(\sqrt{33}+\sqrt{3}\right)\sqrt{33}}{30\times 11}$$

Multiply 30 and 11 to get 330.

$$\frac{\left(11-\sqrt{11}\right)\left(\sqrt{33}+\sqrt{3}\right)\sqrt{33}}{330}$$

Apply the distributive property by multiplying each term of 11-\sqrt{11} by each term of \sqrt{33}+\sqrt{3}.

$$\frac{\left(11\sqrt{33}+11\sqrt{3}-\sqrt{11}\sqrt{33}-\sqrt{11}\sqrt{3}\right)\sqrt{33}}{330}$$

Factor 33=11\times 3. Rewrite the square root of the product \sqrt{11\times 3} as the product of square roots \sqrt{11}\sqrt{3}.

$$\frac{\left(11\sqrt{33}+11\sqrt{3}-\sqrt{11}\sqrt{11}\sqrt{3}-\sqrt{11}\sqrt{3}\right)\sqrt{33}}{330}$$

Multiply \sqrt{11} and \sqrt{11} to get 11.

$$\frac{\left(11\sqrt{33}+11\sqrt{3}-11\sqrt{3}-\sqrt{11}\sqrt{3}\right)\sqrt{33}}{330}$$

Combine 11\sqrt{3} and -11\sqrt{3} to get 0.

$$\frac{\left(11\sqrt{33}-\sqrt{11}\sqrt{3}\right)\sqrt{33}}{330}$$

To multiply \sqrt{11} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\left(11\sqrt{33}-\sqrt{33}\right)\sqrt{33}}{330}$$

Combine 11\sqrt{33} and -\sqrt{33} to get 10\sqrt{33}.

$$\frac{10\sqrt{33}\sqrt{33}}{330}$$

Multiply \sqrt{33} and \sqrt{33} to get 33.

$$\frac{10\times 33}{330}$$

Multiply 10 and 33 to get 330.

$$\frac{330}{330}$$

Divide 330 by 330 to get 1.

$$1$$

Factor

$$1$$