Evaluate
2+4i
Real Part
2
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\frac{3\times 4+3\times \left(-2i\right)+2i\times 4+2\left(-2\right)i^{2}}{2-3i}
Multiply complex numbers 3+2i and 4-2i like you multiply binomials.
\frac{3\times 4+3\times \left(-2i\right)+2i\times 4+2\left(-2\right)\left(-1\right)}{2-3i}
By definition, i^{2} is -1.
\frac{12-6i+8i+4}{2-3i}
Do the multiplications in 3\times 4+3\times \left(-2i\right)+2i\times 4+2\left(-2\right)\left(-1\right).
\frac{12+4+\left(-6+8\right)i}{2-3i}
Combine the real and imaginary parts in 12-6i+8i+4.
\frac{16+2i}{2-3i}
Do the additions in 12+4+\left(-6+8\right)i.
\frac{\left(16+2i\right)\left(2+3i\right)}{\left(2-3i\right)\left(2+3i\right)}
Multiply both numerator and denominator by the complex conjugate of the denominator, 2+3i.
\frac{\left(16+2i\right)\left(2+3i\right)}{2^{2}-3^{2}i^{2}}
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(16+2i\right)\left(2+3i\right)}{13}
By definition, i^{2} is -1. Calculate the denominator.
\frac{16\times 2+16\times \left(3i\right)+2i\times 2+2\times 3i^{2}}{13}
Multiply complex numbers 16+2i and 2+3i like you multiply binomials.
\frac{16\times 2+16\times \left(3i\right)+2i\times 2+2\times 3\left(-1\right)}{13}
By definition, i^{2} is -1.
\frac{32+48i+4i-6}{13}
Do the multiplications in 16\times 2+16\times \left(3i\right)+2i\times 2+2\times 3\left(-1\right).
\frac{32-6+\left(48+4\right)i}{13}
Combine the real and imaginary parts in 32+48i+4i-6.
\frac{26+52i}{13}
Do the additions in 32-6+\left(48+4\right)i.
2+4i
Divide 26+52i by 13 to get 2+4i.
Re(\frac{3\times 4+3\times \left(-2i\right)+2i\times 4+2\left(-2\right)i^{2}}{2-3i})
Multiply complex numbers 3+2i and 4-2i like you multiply binomials.
Re(\frac{3\times 4+3\times \left(-2i\right)+2i\times 4+2\left(-2\right)\left(-1\right)}{2-3i})
By definition, i^{2} is -1.
Re(\frac{12-6i+8i+4}{2-3i})
Do the multiplications in 3\times 4+3\times \left(-2i\right)+2i\times 4+2\left(-2\right)\left(-1\right).
Re(\frac{12+4+\left(-6+8\right)i}{2-3i})
Combine the real and imaginary parts in 12-6i+8i+4.
Re(\frac{16+2i}{2-3i})
Do the additions in 12+4+\left(-6+8\right)i.
Re(\frac{\left(16+2i\right)\left(2+3i\right)}{\left(2-3i\right)\left(2+3i\right)})
Multiply both numerator and denominator of \frac{16+2i}{2-3i} by the complex conjugate of the denominator, 2+3i.
Re(\frac{\left(16+2i\right)\left(2+3i\right)}{2^{2}-3^{2}i^{2}})
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{\left(16+2i\right)\left(2+3i\right)}{13})
By definition, i^{2} is -1. Calculate the denominator.
Re(\frac{16\times 2+16\times \left(3i\right)+2i\times 2+2\times 3i^{2}}{13})
Multiply complex numbers 16+2i and 2+3i like you multiply binomials.
Re(\frac{16\times 2+16\times \left(3i\right)+2i\times 2+2\times 3\left(-1\right)}{13})
By definition, i^{2} is -1.
Re(\frac{32+48i+4i-6}{13})
Do the multiplications in 16\times 2+16\times \left(3i\right)+2i\times 2+2\times 3\left(-1\right).
Re(\frac{32-6+\left(48+4\right)i}{13})
Combine the real and imaginary parts in 32+48i+4i-6.
Re(\frac{26+52i}{13})
Do the additions in 32-6+\left(48+4\right)i.
Re(2+4i)
Divide 26+52i by 13 to get 2+4i.
2
The real part of 2+4i is 2.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}