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

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

Evaluate

$$\frac{2\sqrt{7}+21}{59}\approx 0.445618689$$

Solution Steps

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

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

Consider \left(3\sqrt{7}-2\right)\left(3\sqrt{7}+2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(3\sqrt{7}\right)^{2}.

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

Calculate 3 to the power of 2 and get 9.

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

The square of \sqrt{7} is 7.

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

Multiply 9 and 7 to get 63.

$$\frac{\sqrt{7}\left(3\sqrt{7}+2\right)}{63-2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\sqrt{7}\left(3\sqrt{7}+2\right)}{63-4}$$

Subtract 4 from 63 to get 59.

$$\frac{\sqrt{7}\left(3\sqrt{7}+2\right)}{59}$$

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

$$\frac{3\left(\sqrt{7}\right)^{2}+2\sqrt{7}}{59}$$

The square of \sqrt{7} is 7.

$$\frac{3\times 7+2\sqrt{7}}{59}$$

Multiply 3 and 7 to get 21.

$$\frac{21+2\sqrt{7}}{59}$$