Solve frac{3sqrt{3}-1}{3sqrt{2}-1}

$$\frac{ 3 \sqrt{ 3 } -1 }{ 3 \sqrt{ 2 } -1 }$$

Evaluate

$$\frac{3\sqrt{3}+9\sqrt{6}-3\sqrt{2}-1}{17}\approx 1.294054084$$

Solution Steps

Rationalize the denominator of \frac{3\sqrt{3}-1}{3\sqrt{2}-1} by multiplying numerator and denominator by 3\sqrt{2}+1.

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

Consider \left(3\sqrt{2}-1\right)\left(3\sqrt{2}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate 3 to the power of 2 and get 9.

$$\frac{\left(3\sqrt{3}-1\right)\left(3\sqrt{2}+1\right)}{9\left(\sqrt{2}\right)^{2}-1^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(3\sqrt{3}-1\right)\left(3\sqrt{2}+1\right)}{9\times 2-1^{2}}$$

Multiply 9 and 2 to get 18.

$$\frac{\left(3\sqrt{3}-1\right)\left(3\sqrt{2}+1\right)}{18-1^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{\left(3\sqrt{3}-1\right)\left(3\sqrt{2}+1\right)}{18-1}$$

Subtract 1 from 18 to get 17.

$$\frac{\left(3\sqrt{3}-1\right)\left(3\sqrt{2}+1\right)}{17}$$

Apply the distributive property by multiplying each term of 3\sqrt{3}-1 by each term of 3\sqrt{2}+1.

$$\frac{9\sqrt{3}\sqrt{2}+3\sqrt{3}-3\sqrt{2}-1}{17}$$

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

$$\frac{9\sqrt{6}+3\sqrt{3}-3\sqrt{2}-1}{17}$$