Solve 2/3+sqrt{-5}:3

$$\frac { 2 } { 3 + \sqrt { - 5 } } : 3$$

Evaluate (complex solution)

$$-\frac{\sqrt{5}i}{21}+\frac{1}{7}\approx 0.142857143-0.106479427i$$

Solution Steps

Express \frac{\frac{2}{3+\sqrt{-5}}}{3} as a single fraction.

$$\frac{2}{\left(3+\sqrt{-5}\right)\times 3}$$

Factor -5=5\left(-1\right). Rewrite the square root of the product \sqrt{5\left(-1\right)} as the product of square roots \sqrt{5}\sqrt{-1}. By definition, the square root of -1 is i.

$$\frac{2}{\left(3+\sqrt{5}i\right)\times 3}$$

Use the distributive property to multiply 3+\sqrt{5}i by 3.

$$\frac{2}{9+3\sqrt{5}i}$$

Multiply 3 and i to get 3i.

$$\frac{2}{9+3i\sqrt{5}}$$

Rationalize the denominator of \frac{2}{9+3i\sqrt{5}} by multiplying numerator and denominator by 9-3i\sqrt{5}.

$$\frac{2\left(9-3i\sqrt{5}\right)}{\left(9+3i\sqrt{5}\right)\left(9-3i\sqrt{5}\right)}$$

Consider \left(9+3i\sqrt{5}\right)\left(9-3i\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\left(9-3i\sqrt{5}\right)}{9^{2}-\left(3i\sqrt{5}\right)^{2}}$$

Calculate 9 to the power of 2 and get 81.

$$\frac{2\left(9-3i\sqrt{5}\right)}{81-\left(3i\sqrt{5}\right)^{2}}$$

Expand \left(3i\sqrt{5}\right)^{2}.

$$\frac{2\left(9-3i\sqrt{5}\right)}{81-\left(3i\right)^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate 3i to the power of 2 and get -9.

$$\frac{2\left(9-3i\sqrt{5}\right)}{81-\left(-9\left(\sqrt{5}\right)^{2}\right)}$$

The square of \sqrt{5} is 5.

$$\frac{2\left(9-3i\sqrt{5}\right)}{81-\left(-9\times 5\right)}$$

Multiply -9 and 5 to get -45.

$$\frac{2\left(9-3i\sqrt{5}\right)}{81-\left(-45\right)}$$

Multiply -1 and -45 to get 45.

$$\frac{2\left(9-3i\sqrt{5}\right)}{81+45}$$

Add 81 and 45 to get 126.

$$\frac{2\left(9-3i\sqrt{5}\right)}{126}$$

Divide 2\left(9-3i\sqrt{5}\right) by 126 to get \frac{1}{63}\left(9-3i\sqrt{5}\right).

$$\frac{1}{63}\left(9-3i\sqrt{5}\right)$$

Use the distributive property to multiply \frac{1}{63} by 9-3i\sqrt{5}.

$$\frac{1}{63}\times 9+\frac{1}{63}\times \left(-3i\right)\sqrt{5}$$

Multiply \frac{1}{63} and 9 to get \frac{9}{63}.

$$\frac{9}{63}+\frac{1}{63}\times \left(-3i\right)\sqrt{5}$$

Reduce the fraction \frac{9}{63} to lowest terms by extracting and canceling out 9.

$$\frac{1}{7}+\frac{1}{63}\times \left(-3i\right)\sqrt{5}$$

Multiply \frac{1}{63} and -3i to get -\frac{1}{21}i.

$$\frac{1}{7}-\frac{1}{21}i\sqrt{5}$$

Evaluate

$$\text{Indeterminate}$$

Solution Steps

Express \frac{\frac{2}{3+\sqrt{-5}}}{3} as a single fraction.

$$\frac{2}{\left(3+\sqrt{-5}\right)\times 3}$$

Use the distributive property to multiply 3+\sqrt{-5} by 3.

$$\frac{2}{9+3\sqrt{-5}}$$

Rationalize the denominator of \frac{2}{9+3\sqrt{-5}} by multiplying numerator and denominator by 9-3\sqrt{-5}.

$$\frac{2\left(9-3\sqrt{-5}\right)}{\left(9+3\sqrt{-5}\right)\left(9-3\sqrt{-5}\right)}$$

Consider \left(9+3\sqrt{-5}\right)\left(9-3\sqrt{-5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\left(9-3\sqrt{-5}\right)}{9^{2}-\left(3\sqrt{-5}\right)^{2}}$$

Calculate 9 to the power of 2 and get 81.

$$\frac{2\left(9-3\sqrt{-5}\right)}{81-\left(3\sqrt{-5}\right)^{2}}$$

Expand \left(3\sqrt{-5}\right)^{2}.

$$\frac{2\left(9-3\sqrt{-5}\right)}{81-3^{2}\left(\sqrt{-5}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{2\left(9-3\sqrt{-5}\right)}{81-9\left(\sqrt{-5}\right)^{2}}$$

Calculate \sqrt{-5} to the power of 2 and get -5.

$$\frac{2\left(9-3\sqrt{-5}\right)}{81-9\left(-5\right)}$$

Multiply 9 and -5 to get -45.

$$\frac{2\left(9-3\sqrt{-5}\right)}{81-\left(-45\right)}$$

Multiply -1 and -45 to get 45.

$$\frac{2\left(9-3\sqrt{-5}\right)}{81+45}$$

Add 81 and 45 to get 126.

$$\frac{2\left(9-3\sqrt{-5}\right)}{126}$$

Divide 2\left(9-3\sqrt{-5}\right) by 126 to get \frac{1}{63}\left(9-3\sqrt{-5}\right).

$$\frac{1}{63}\left(9-3\sqrt{-5}\right)$$

Use the distributive property to multiply \frac{1}{63} by 9-3\sqrt{-5}.

$$\frac{1}{63}\times 9+\frac{1}{63}\left(-3\right)\sqrt{-5}$$

Multiply \frac{1}{63} and 9 to get \frac{9}{63}.

$$\frac{9}{63}+\frac{1}{63}\left(-3\right)\sqrt{-5}$$

Reduce the fraction \frac{9}{63} to lowest terms by extracting and canceling out 9.

$$\frac{1}{7}+\frac{1}{63}\left(-3\right)\sqrt{-5}$$

Multiply \frac{1}{63} and -3 to get \frac{-3}{63}.

$$\frac{1}{7}+\frac{-3}{63}\sqrt{-5}$$

Reduce the fraction \frac{-3}{63} to lowest terms by extracting and canceling out 3.

$$\frac{1}{7}-\frac{1}{21}\sqrt{-5}$$

Real Part (complex solution)

$$\frac{1}{7} = 0.14285714285714285$$