Solve frac{-3-2sqrt{-1}}{-7-6sqrt{-1}}

$$\frac { - 3 - 2 \sqrt { - 1 } } { - 7 - 6 \sqrt { - 1 } }$$

Evaluate

$$\text{Indeterminate}$$

Solution Steps

Rationalize the denominator of \frac{-3-2\sqrt{-1}}{-7-6\sqrt{-1}} by multiplying numerator and denominator by -7+6\sqrt{-1}.

$$\frac{\left(-3-2\sqrt{-1}\right)\left(-7+6\sqrt{-1}\right)}{\left(-7-6\sqrt{-1}\right)\left(-7+6\sqrt{-1}\right)}$$

Consider \left(-7-6\sqrt{-1}\right)\left(-7+6\sqrt{-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}.

$$\frac{\left(-3-2\sqrt{-1}\right)\left(-7+6\sqrt{-1}\right)}{\left(-7\right)^{2}-\left(-6\sqrt{-1}\right)^{2}}$$

Calculate -7 to the power of 2 and get 49.

$$\frac{\left(-3-2\sqrt{-1}\right)\left(-7+6\sqrt{-1}\right)}{49-\left(-6\sqrt{-1}\right)^{2}}$$

Expand \left(-6\sqrt{-1}\right)^{2}.

$$\frac{\left(-3-2\sqrt{-1}\right)\left(-7+6\sqrt{-1}\right)}{49-\left(-6\right)^{2}\left(\sqrt{-1}\right)^{2}}$$

Calculate -6 to the power of 2 and get 36.

$$\frac{\left(-3-2\sqrt{-1}\right)\left(-7+6\sqrt{-1}\right)}{49-36\left(\sqrt{-1}\right)^{2}}$$

Calculate \sqrt{-1} to the power of 2 and get -1.

$$\frac{\left(-3-2\sqrt{-1}\right)\left(-7+6\sqrt{-1}\right)}{49-36\left(-1\right)}$$

Multiply 36 and -1 to get -36.

$$\frac{\left(-3-2\sqrt{-1}\right)\left(-7+6\sqrt{-1}\right)}{49-\left(-36\right)}$$

Multiply -1 and -36 to get 36.

$$\frac{\left(-3-2\sqrt{-1}\right)\left(-7+6\sqrt{-1}\right)}{49+36}$$

Add 49 and 36 to get 85.

$$\frac{\left(-3-2\sqrt{-1}\right)\left(-7+6\sqrt{-1}\right)}{85}$$

Apply the distributive property by multiplying each term of -3-2\sqrt{-1} by each term of -7+6\sqrt{-1}.

$$\frac{21-18\sqrt{-1}+14\sqrt{-1}-12\left(\sqrt{-1}\right)^{2}}{85}$$

Combine -18\sqrt{-1} and 14\sqrt{-1} to get -4\sqrt{-1}.

$$\frac{21-4\sqrt{-1}-12\left(\sqrt{-1}\right)^{2}}{85}$$

Calculate \sqrt{-1} to the power of 2 and get -1.

$$\frac{21-4\sqrt{-1}-12\left(-1\right)}{85}$$

Multiply -12 and -1 to get 12.

$$\frac{21-4\sqrt{-1}+12}{85}$$

Add 21 and 12 to get 33.

$$\frac{33-4\sqrt{-1}}{85}$$