Examples
−53+2i
Step 1
Multiply the numerator and denominator of −53+2i by the conjugate of 3+2i to make the denominator real.
−53+2i⋅3−2i3−2i
Step 2
Step 2.1
Combine.
−5(3−2i)(3+2i)(3−2i)
Step 2.2
Simplify the numerator.
Step 2.2.1
Apply the distributive property.
−5⋅3−5(−2i)(3+2i)(3−2i)
Step 2.2.2
Multiply −5 by 3.
−15−5(−2i)(3+2i)(3−2i)
Step 2.2.3
Multiply −2 by −5.
−15+10i(3+2i)(3−2i)
−15+10i(3+2i)(3−2i)
Step 2.3
Simplify the denominator.
Step 2.3.1
Expand (3+2i)(3−2i) using the FOIL Method.
Step 2.3.1.1
Apply the distributive property.
−15+10i3(3−2i)+2i(3−2i)
Step 2.3.1.2
Apply the distributive property.
−15+10i3⋅3+3(−2i)+2i(3−2i)
Step 2.3.1.3
Apply the distributive property.
−15+10i3⋅3+3(−2i)+2i⋅3+2i(−2i)
−15+10i3⋅3+3(−2i)+2i⋅3+2i(−2i)
Step 2.3.2
Simplify.
Step 2.3.2.1
Multiply 3 by 3.
−15+10i9+3(−2i)+2i⋅3+2i(−2i)
Step 2.3.2.2
Multiply −2 by 3.
−15+10i9−6i+2i⋅3+2i(−2i)
Step 2.3.2.3
Multiply 3 by 2.
−15+10i9−6i+6i+2i(−2i)
Step 2.3.2.4
Multiply −2 by 2.
−15+10i9−6i+6i−4ii
Step 2.3.2.5
Raise i to the power of 1.
−15+10i9−6i+6i−4(i1i)
Step 2.3.2.6
Raise i to the power of 1.
−15+10i9−6i+6i−4(i1i1)
Step 2.3.2.7
Use the power rule aman=am+n to combine exponents.
−15+10i9−6i+6i−4i1+1
Step 2.3.2.8
Add 1 and 1.
−15+10i9−6i+6i−4i2
Step 2.3.2.9
Add −6i and 6i.
−15+10i9+0−4i2
Step 2.3.2.10
Add 9 and 0.
−15+10i9−4i2
−15+10i9−4i2
Step 2.3.3
Simplify each term.
Step 2.3.3.1
Rewrite i2 as −1.
−15+10i9−4⋅−1
Step 2.3.3.2
Multiply −4 by −1.
−15+10i9+4
−15+10i9+4
Step 2.3.4
Add 9 and 4.
−15+10i13
−15+10i13
−15+10i13
Step 3
Split the fraction −15+10i13 into two fractions.
−1513+10i13
Step 4
Move the negative in front of the fraction.
−1513+10i13
