Solve frac{3sqrt{2}}{7sqrt{2}-6sqrt{3}}

$$\frac{ 3 \sqrt{ 2 } }{ 7 \sqrt{ 2 } -6 \sqrt{ 3 } }$$

Evaluate

$$\frac{-9\sqrt{6}-21}{5}\approx -8.609081537$$

Solution Steps

Rationalize the denominator of \frac{3\sqrt{2}}{7\sqrt{2}-6\sqrt{3}} by multiplying numerator and denominator by 7\sqrt{2}+6\sqrt{3}.

$$\frac{3\sqrt{2}\left(7\sqrt{2}+6\sqrt{3}\right)}{\left(7\sqrt{2}-6\sqrt{3}\right)\left(7\sqrt{2}+6\sqrt{3}\right)}$$

Consider \left(7\sqrt{2}-6\sqrt{3}\right)\left(7\sqrt{2}+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}.

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

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

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

Calculate 7 to the power of 2 and get 49.

$$\frac{3\sqrt{2}\left(7\sqrt{2}+6\sqrt{3}\right)}{49\left(\sqrt{2}\right)^{2}-\left(-6\sqrt{3}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{3\sqrt{2}\left(7\sqrt{2}+6\sqrt{3}\right)}{49\times 2-\left(-6\sqrt{3}\right)^{2}}$$

Multiply 49 and 2 to get 98.

$$\frac{3\sqrt{2}\left(7\sqrt{2}+6\sqrt{3}\right)}{98-\left(-6\sqrt{3}\right)^{2}}$$

Expand \left(-6\sqrt{3}\right)^{2}.

$$\frac{3\sqrt{2}\left(7\sqrt{2}+6\sqrt{3}\right)}{98-\left(-6\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate -6 to the power of 2 and get 36.

$$\frac{3\sqrt{2}\left(7\sqrt{2}+6\sqrt{3}\right)}{98-36\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{3\sqrt{2}\left(7\sqrt{2}+6\sqrt{3}\right)}{98-36\times 3}$$

Multiply 36 and 3 to get 108.

$$\frac{3\sqrt{2}\left(7\sqrt{2}+6\sqrt{3}\right)}{98-108}$$

Subtract 108 from 98 to get -10.

$$\frac{3\sqrt{2}\left(7\sqrt{2}+6\sqrt{3}\right)}{-10}$$

Use the distributive property to multiply 3\sqrt{2} by 7\sqrt{2}+6\sqrt{3}.

$$\frac{21\left(\sqrt{2}\right)^{2}+18\sqrt{3}\sqrt{2}}{-10}$$

The square of \sqrt{2} is 2.

$$\frac{21\times 2+18\sqrt{3}\sqrt{2}}{-10}$$

Multiply 21 and 2 to get 42.

$$\frac{42+18\sqrt{3}\sqrt{2}}{-10}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{42+18\sqrt{6}}{-10}$$