Solve frac{-33.965*sqrt{-1}}{325-33.965sqrt{-1}}

$$\frac{ -33.965 \times \sqrt{ -1 } }{ 325-33.965 \sqrt{ -1 } }$$

Evaluate (complex solution)

$$\frac{46144849}{4271144849}-\frac{441545000}{4271144849}i\approx 0.01080386-0.103378606i$$

Solution Steps

Calculate the square root of -1 and get i.

$$\frac{-33.965i}{325-33.965\sqrt{-1}}$$

Calculate the square root of -1 and get i.

$$\frac{-33.965i}{325-33.965i}$$

Multiply both numerator and denominator by the complex conjugate of the denominator, 325+33.965i.

$$\frac{-33.965i\left(325+33.965i\right)}{\left(325-33.965i\right)\left(325+33.965i\right)}$$

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

$$\frac{-33.965i\left(325+33.965i\right)}{325^{2}-33.965^{2}i^{2}}$$

By definition, i^{2} is -1. Calculate the denominator.

$$\frac{-33.965i\left(325+33.965i\right)}{106778.621225}$$

Multiply -33.965i times 325+33.965i.

$$\frac{-33.965i\times 325-33.965\times 33.965i^{2}}{106778.621225}$$

By definition, i^{2} is -1.

$$\frac{-33.965i\times 325-33.965\times 33.965\left(-1\right)}{106778.621225}$$

Do the multiplications in -33.965i\times 325-33.965\times 33.965\left(-1\right). Reorder the terms.

$$\frac{1153.621225-11038.625i}{106778.621225}$$

Divide 1153.621225-11038.625i by 106778.621225 to get \frac{46144849}{4271144849}-\frac{441545000}{4271144849}i.

$$\frac{46144849}{4271144849}-\frac{441545000}{4271144849}i$$

Real Part (complex solution)

$$\frac{46144849}{4271144849} = 0.01080385953447677$$

Solution Steps

Calculate the square root of -1 and get i.

$$Re(\frac{-33.965i}{325-33.965\sqrt{-1}})$$

Calculate the square root of -1 and get i.

$$Re(\frac{-33.965i}{325-33.965i})$$

Multiply both numerator and denominator of \frac{-33.965i}{325-33.965i} by the complex conjugate of the denominator, 325+33.965i.

$$Re(\frac{-33.965i\left(325+33.965i\right)}{\left(325-33.965i\right)\left(325+33.965i\right)})$$

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

$$Re(\frac{-33.965i\left(325+33.965i\right)}{325^{2}-33.965^{2}i^{2}})$$

By definition, i^{2} is -1. Calculate the denominator.

$$Re(\frac{-33.965i\left(325+33.965i\right)}{106778.621225})$$

Multiply -33.965i times 325+33.965i.

$$Re(\frac{-33.965i\times 325-33.965\times 33.965i^{2}}{106778.621225})$$

By definition, i^{2} is -1.

$$Re(\frac{-33.965i\times 325-33.965\times 33.965\left(-1\right)}{106778.621225})$$

Do the multiplications in -33.965i\times 325-33.965\times 33.965\left(-1\right). Reorder the terms.

$$Re(\frac{1153.621225-11038.625i}{106778.621225})$$

Divide 1153.621225-11038.625i by 106778.621225 to get \frac{46144849}{4271144849}-\frac{441545000}{4271144849}i.

$$Re(\frac{46144849}{4271144849}-\frac{441545000}{4271144849}i)$$

The real part of \frac{46144849}{4271144849}-\frac{441545000}{4271144849}i is \frac{46144849}{4271144849}.

$$\frac{46144849}{4271144849}$$

Evaluate

$$\text{Indeterminate}$$

Solution Steps

Rationalize the denominator of \frac{-33.965\sqrt{-1}}{325-33.965\sqrt{-1}} by multiplying numerator and denominator by 325+33.965\sqrt{-1}.

$$\frac{-33.965\sqrt{-1}\left(325+33.965\sqrt{-1}\right)}{\left(325-33.965\sqrt{-1}\right)\left(325+33.965\sqrt{-1}\right)}$$

Consider \left(325-33.965\sqrt{-1}\right)\left(325+33.965\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}.

$$\frac{-33.965\sqrt{-1}\left(325+33.965\sqrt{-1}\right)}{325^{2}-\left(-33.965\sqrt{-1}\right)^{2}}$$

Calculate 325 to the power of 2 and get 105625.

$$\frac{-33.965\sqrt{-1}\left(325+33.965\sqrt{-1}\right)}{105625-\left(-33.965\sqrt{-1}\right)^{2}}$$

Expand \left(-33.965\sqrt{-1}\right)^{2}.

$$\frac{-33.965\sqrt{-1}\left(325+33.965\sqrt{-1}\right)}{105625-\left(-33.965\right)^{2}\left(\sqrt{-1}\right)^{2}}$$

Calculate -33.965 to the power of 2 and get 1153.621225.

$$\frac{-33.965\sqrt{-1}\left(325+33.965\sqrt{-1}\right)}{105625-1153.621225\left(\sqrt{-1}\right)^{2}}$$

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

$$\frac{-33.965\sqrt{-1}\left(325+33.965\sqrt{-1}\right)}{105625-1153.621225\left(-1\right)}$$

Multiply 1153.621225 and -1 to get -1153.621225.

$$\frac{-33.965\sqrt{-1}\left(325+33.965\sqrt{-1}\right)}{105625-\left(-1153.621225\right)}$$

Multiply -1 and -1153.621225 to get 1153.621225.

$$\frac{-33.965\sqrt{-1}\left(325+33.965\sqrt{-1}\right)}{105625+1153.621225}$$

Add 105625 and 1153.621225 to get 106778.621225.

$$\frac{-33.965\sqrt{-1}\left(325+33.965\sqrt{-1}\right)}{106778.621225}$$

Divide -33.965\sqrt{-1}\left(325+33.965\sqrt{-1}\right) by 106778.621225 to get -\frac{1358600}{4271144849}\sqrt{-1}\left(325+33.965\sqrt{-1}\right).

$$-\frac{1358600}{4271144849}\sqrt{-1}\left(325+33.965\sqrt{-1}\right)$$

Use the distributive property to multiply -\frac{1358600}{4271144849}\sqrt{-1} by 325+33.965\sqrt{-1}.

$$-\frac{441545000}{4271144849}\sqrt{-1}-\frac{46144849}{4271144849}\left(\sqrt{-1}\right)^{2}$$

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

$$-\frac{441545000}{4271144849}\sqrt{-1}-\frac{46144849}{4271144849}\left(-1\right)$$

Multiply -\frac{46144849}{4271144849} and -1 to get \frac{46144849}{4271144849}.

$$-\frac{441545000}{4271144849}\sqrt{-1}+\frac{46144849}{4271144849}$$