Solve -7/-2sqrt{7-7}

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

Evaluate

$$\frac{7-2\sqrt{7}}{3}\approx 0.569499126$$

Solution Steps

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

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

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

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

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

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

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

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

The square of \sqrt{7} is 7.

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

Multiply 4 and 7 to get 28.

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

Calculate 7 to the power of 2 and get 49.

$$\frac{-7\left(-2\sqrt{7}+7\right)}{28-49}$$

Subtract 49 from 28 to get -21.

$$\frac{-7\left(-2\sqrt{7}+7\right)}{-21}$$

Divide -7\left(-2\sqrt{7}+7\right) by -21 to get \frac{1}{3}\left(-2\sqrt{7}+7\right).

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

Use the distributive property to multiply \frac{1}{3} by -2\sqrt{7}+7.

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

Multiply \frac{1}{3} and -2 to get \frac{-2}{3}.

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

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

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

Multiply \frac{1}{3} and 7 to get \frac{7}{3}.

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