Solve frac{3sqrt{3}-2}{2sqrt{7}+1}

$$\frac { 3 \sqrt { 3 } - 2 } { 2 \sqrt { 7 } + 1 }$$

Evaluate

$$\frac{2\sqrt{21}}{9}-\frac{\sqrt{3}}{9}-\frac{4\sqrt{7}}{27}+\frac{2}{27}\approx 0.508010982$$

Solution Steps

Rationalize the denominator of \frac{3\sqrt{3}-2}{2\sqrt{7}+1} by multiplying numerator and denominator by 2\sqrt{7}-1.

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

Consider \left(2\sqrt{7}+1\right)\left(2\sqrt{7}-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}.

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

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

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

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(3\sqrt{3}-2\right)\left(2\sqrt{7}-1\right)}{4\left(\sqrt{7}\right)^{2}-1^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{\left(3\sqrt{3}-2\right)\left(2\sqrt{7}-1\right)}{4\times 7-1^{2}}$$

Multiply 4 and 7 to get 28.

$$\frac{\left(3\sqrt{3}-2\right)\left(2\sqrt{7}-1\right)}{28-1^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{\left(3\sqrt{3}-2\right)\left(2\sqrt{7}-1\right)}{28-1}$$

Subtract 1 from 28 to get 27.

$$\frac{\left(3\sqrt{3}-2\right)\left(2\sqrt{7}-1\right)}{27}$$

Apply the distributive property by multiplying each term of 3\sqrt{3}-2 by each term of 2\sqrt{7}-1.

$$\frac{6\sqrt{3}\sqrt{7}-3\sqrt{3}-4\sqrt{7}+2}{27}$$

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

$$\frac{6\sqrt{21}-3\sqrt{3}-4\sqrt{7}+2}{27}$$