Solve 240/2.5+2.5sqrt{3i+10+sqrt{300}i}

$$\frac{ 240 }{ 2.5+2.5 \sqrt{ 3 } i+10+ \sqrt{ 300 } i }$$

Evaluate

$$\frac{2\times 240}{25+i\times 25\sqrt{3}}\approx 4.8-8.313843876i$$

Solution Steps

Add 2.5 and 10 to get 12.5.

$$\frac{240}{12.5+2.5i\sqrt{3}+i\sqrt{300}}$$

Factor 300=10^{2}\times 3. Rewrite the square root of the product \sqrt{10^{2}\times 3} as the product of square roots \sqrt{10^{2}}\sqrt{3}. Take the square root of 10^{2}.

$$\frac{240}{12.5+2.5i\sqrt{3}+i\times 10\sqrt{3}}$$

Combine 2.5i\sqrt{3} and 10i\sqrt{3} to get 12.5i\sqrt{3}.

$$\frac{240}{12.5+12.5i\sqrt{3}}$$

Rationalize the denominator of \frac{240}{12.5+12.5i\sqrt{3}} by multiplying numerator and denominator by 12.5-12.5i\sqrt{3}.

$$\frac{240\left(12.5-12.5i\sqrt{3}\right)}{\left(12.5+12.5i\sqrt{3}\right)\left(12.5-12.5i\sqrt{3}\right)}$$

Consider \left(12.5+12.5i\sqrt{3}\right)\left(12.5-12.5i\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{240\left(12.5-12.5i\sqrt{3}\right)}{12.5^{2}-\left(12.5i\sqrt{3}\right)^{2}}$$

Calculate 12.5 to the power of 2 and get 156.25.

$$\frac{240\left(12.5-12.5i\sqrt{3}\right)}{156.25-\left(12.5i\sqrt{3}\right)^{2}}$$

Expand \left(12.5i\sqrt{3}\right)^{2}.

$$\frac{240\left(12.5-12.5i\sqrt{3}\right)}{156.25-\left(12.5i\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate 12.5i to the power of 2 and get -156.25.

$$\frac{240\left(12.5-12.5i\sqrt{3}\right)}{156.25-\left(-156.25\left(\sqrt{3}\right)^{2}\right)}$$

The square of \sqrt{3} is 3.

$$\frac{240\left(12.5-12.5i\sqrt{3}\right)}{156.25-\left(-156.25\times 3\right)}$$

Multiply -156.25 and 3 to get -468.75.

$$\frac{240\left(12.5-12.5i\sqrt{3}\right)}{156.25-\left(-468.75\right)}$$

Multiply -1 and -468.75 to get 468.75.

$$\frac{240\left(12.5-12.5i\sqrt{3}\right)}{156.25+468.75}$$

Add 156.25 and 468.75 to get 625.

$$\frac{240\left(12.5-12.5i\sqrt{3}\right)}{625}$$

Divide 240\left(12.5-12.5i\sqrt{3}\right) by 625 to get \frac{48}{125}\left(12.5-12.5i\sqrt{3}\right).

$$\frac{48}{125}\left(12.5-12.5i\sqrt{3}\right)$$

Use the distributive property to multiply \frac{48}{125} by 12.5-12.5i\sqrt{3}.

$$\frac{48}{125}\times 12.5+\frac{48}{125}\times \left(-12.5i\right)\sqrt{3}$$

Convert decimal number 12.5 to fraction \frac{125}{10}. Reduce the fraction \frac{125}{10} to lowest terms by extracting and canceling out 5.

$$\frac{48}{125}\times \frac{25}{2}+\frac{48}{125}\times \left(-12.5i\right)\sqrt{3}$$

Multiply \frac{48}{125} times \frac{25}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{48\times 25}{125\times 2}+\frac{48}{125}\times \left(-12.5i\right)\sqrt{3}$$

Do the multiplications in the fraction \frac{48\times 25}{125\times 2}.

$$\frac{1200}{250}+\frac{48}{125}\times \left(-12.5i\right)\sqrt{3}$$

Reduce the fraction \frac{1200}{250} to lowest terms by extracting and canceling out 50.

$$\frac{24}{5}+\frac{48}{125}\times \left(-12.5i\right)\sqrt{3}$$

Multiply \frac{48}{125} and -12.5i to get -\frac{24}{5}i.

$$\frac{24}{5}-\frac{24}{5}i\sqrt{3}$$

Real Part

$$240Re(\frac{2}{25+i\times 25\sqrt{3}})$$