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

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

Evaluate

$$6\sqrt{35}+9\sqrt{15}-2\sqrt{21}-9\approx 52.188177425$$

Solution Steps

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

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

Consider \left(2\sqrt{7}-3\sqrt{3}\right)\left(2\sqrt{7}+3\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}.

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

Expand \left(2\sqrt{7}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{7} is 7.

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

Multiply 4 and 7 to get 28.

$$\frac{\left(3\sqrt{5}-\sqrt{3}\right)\left(2\sqrt{7}+3\sqrt{3}\right)}{28-\left(-3\sqrt{3}\right)^{2}}$$

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

$$\frac{\left(3\sqrt{5}-\sqrt{3}\right)\left(2\sqrt{7}+3\sqrt{3}\right)}{28-\left(-3\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate -3 to the power of 2 and get 9.

$$\frac{\left(3\sqrt{5}-\sqrt{3}\right)\left(2\sqrt{7}+3\sqrt{3}\right)}{28-9\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(3\sqrt{5}-\sqrt{3}\right)\left(2\sqrt{7}+3\sqrt{3}\right)}{28-9\times 3}$$

Multiply 9 and 3 to get 27.

$$\frac{\left(3\sqrt{5}-\sqrt{3}\right)\left(2\sqrt{7}+3\sqrt{3}\right)}{28-27}$$

Subtract 27 from 28 to get 1.

$$\frac{\left(3\sqrt{5}-\sqrt{3}\right)\left(2\sqrt{7}+3\sqrt{3}\right)}{1}$$

Anything divided by one gives itself.

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

Apply the distributive property by multiplying each term of 3\sqrt{5}-\sqrt{3} by each term of 2\sqrt{7}+3\sqrt{3}.

$$6\sqrt{5}\sqrt{7}+9\sqrt{3}\sqrt{5}-2\sqrt{3}\sqrt{7}-3\left(\sqrt{3}\right)^{2}$$

To multiply \sqrt{5} and \sqrt{7}, multiply the numbers under the square root.

$$6\sqrt{35}+9\sqrt{3}\sqrt{5}-2\sqrt{3}\sqrt{7}-3\left(\sqrt{3}\right)^{2}$$

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

$$6\sqrt{35}+9\sqrt{15}-2\sqrt{3}\sqrt{7}-3\left(\sqrt{3}\right)^{2}$$

To multiply \sqrt{3} and \sqrt{7}, multiply the numbers under the square root.

$$6\sqrt{35}+9\sqrt{15}-2\sqrt{21}-3\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$6\sqrt{35}+9\sqrt{15}-2\sqrt{21}-3\times 3$$

Multiply -3 and 3 to get -9.

$$6\sqrt{35}+9\sqrt{15}-2\sqrt{21}-9$$

Factor

$$6 \sqrt{35} + 9 \sqrt{15} - 2 \sqrt{21} - 9 = 52.188177425$$