Solve frac{42sqrt{2}+13}{3sqrt{2}-2sqrt{3}}

$$\frac { 42 \sqrt { 2 } + 13 } { 3 \sqrt { 2 } - 2 \sqrt { 3 } }$$

Evaluate

$$\frac{13\sqrt{2}}{2}+\frac{13\sqrt{3}}{3}+14\sqrt{6}+42\approx 92.990798054$$

Solution Steps

Rationalize the denominator of \frac{42\sqrt{2}+13}{3\sqrt{2}-2\sqrt{3}} by multiplying numerator and denominator by 3\sqrt{2}+2\sqrt{3}.

$$\frac{\left(42\sqrt{2}+13\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{\left(3\sqrt{2}-2\sqrt{3}\right)\left(3\sqrt{2}+2\sqrt{3}\right)}$$

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

$$\frac{\left(42\sqrt{2}+13\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{\left(3\sqrt{2}\right)^{2}-\left(-2\sqrt{3}\right)^{2}}$$

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

$$\frac{\left(42\sqrt{2}+13\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{3^{2}\left(\sqrt{2}\right)^{2}-\left(-2\sqrt{3}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\left(42\sqrt{2}+13\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{9\left(\sqrt{2}\right)^{2}-\left(-2\sqrt{3}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(42\sqrt{2}+13\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{9\times 2-\left(-2\sqrt{3}\right)^{2}}$$

Multiply 9 and 2 to get 18.

$$\frac{\left(42\sqrt{2}+13\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{18-\left(-2\sqrt{3}\right)^{2}}$$

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

$$\frac{\left(42\sqrt{2}+13\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{18-\left(-2\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate -2 to the power of 2 and get 4.

$$\frac{\left(42\sqrt{2}+13\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{18-4\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(42\sqrt{2}+13\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{18-4\times 3}$$

Multiply 4 and 3 to get 12.

$$\frac{\left(42\sqrt{2}+13\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{18-12}$$

Subtract 12 from 18 to get 6.

$$\frac{\left(42\sqrt{2}+13\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{6}$$

Apply the distributive property by multiplying each term of 42\sqrt{2}+13 by each term of 3\sqrt{2}+2\sqrt{3}.

$$\frac{126\left(\sqrt{2}\right)^{2}+84\sqrt{3}\sqrt{2}+39\sqrt{2}+26\sqrt{3}}{6}$$

The square of \sqrt{2} is 2.

$$\frac{126\times 2+84\sqrt{3}\sqrt{2}+39\sqrt{2}+26\sqrt{3}}{6}$$

Multiply 126 and 2 to get 252.

$$\frac{252+84\sqrt{3}\sqrt{2}+39\sqrt{2}+26\sqrt{3}}{6}$$

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

$$\frac{252+84\sqrt{6}+39\sqrt{2}+26\sqrt{3}}{6}$$