# Simplify (3+i)/(1+i)

3+i1+i
Multiply the numerator and denominator of 3+i1+1i by the conjugate of 1+1i to make the denominator real.
3+i1+1i⋅1-i1-i
Multiply.
Combine.
(3+i)(1-i)(1+1i)(1-i)
Simplify the numerator.
Expand (3+i)(1-i) using the FOIL Method.
Apply the distributive property.
3(1-i)+i(1-i)(1+1i)(1-i)
Apply the distributive property.
3⋅1+3(-i)+i(1-i)(1+1i)(1-i)
Apply the distributive property.
3⋅1+3(-i)+i⋅1+i(-i)(1+1i)(1-i)
3⋅1+3(-i)+i⋅1+i(-i)(1+1i)(1-i)
Simplify and combine like terms.
Simplify each term.
Multiply 3 by 1.
3+3(-i)+i⋅1+i(-i)(1+1i)(1-i)
Multiply -1 by 3.
3-3i+i⋅1+i(-i)(1+1i)(1-i)
Multiply i by 1.
3-3i+i+i(-i)(1+1i)(1-i)
Multiply i(-i).
Raise i to the power of 1.
3-3i+i-(i1i)(1+1i)(1-i)
Raise i to the power of 1.
3-3i+i-(i1i1)(1+1i)(1-i)
Use the power rule aman=am+n to combine exponents.
3-3i+i-i1+1(1+1i)(1-i)
3-3i+i-i2(1+1i)(1-i)
3-3i+i-i2(1+1i)(1-i)
Rewrite i2 as -1.
3-3i+i–1(1+1i)(1-i)
Multiply -1 by -1.
3-3i+i+1(1+1i)(1-i)
3-3i+i+1(1+1i)(1-i)
4-3i+i(1+1i)(1-i)
4-2i(1+1i)(1-i)
4-2i(1+1i)(1-i)
4-2i(1+1i)(1-i)
Simplify the denominator.
Expand (1+1i)(1-i) using the FOIL Method.
Apply the distributive property.
4-2i1(1-i)+1i(1-i)
Apply the distributive property.
4-2i1⋅1+1(-i)+1i(1-i)
Apply the distributive property.
4-2i1⋅1+1(-i)+1i⋅1+1i(-i)
4-2i1⋅1+1(-i)+1i⋅1+1i(-i)
Simplify.
Multiply 1 by 1.
4-2i1+1(-i)+1i⋅1+1i(-i)
Multiply -1 by 1.
4-2i1-1i+1i⋅1+1i(-i)
Multiply 1 by 1.
4-2i1-1i+1i+1i(-i)
Multiply -1 by 1.
4-2i1-1i+1i-ii
Raise i to the power of 1.
4-2i1-1i+1i-(i1i)
Raise i to the power of 1.
4-2i1-1i+1i-(i1i1)
Use the power rule aman=am+n to combine exponents.
4-2i1-1i+1i-i1+1
4-2i1-1i+1i-i2
4-2i1+0-i2
4-2i1-i2
4-2i1-i2
Simplify each term.
Rewrite i2 as -1.
4-2i1–1
Multiply -1 by -1.
4-2i1+1
4-2i1+1
4-2i2
4-2i2
4-2i2
Cancel the common factor of 4-2i and 2.
Factor 2 out of 4.
2⋅2-2i2
Factor 2 out of -2i.
2⋅2+2(-i)2
Factor 2 out of 2(2)+2(-i).
2(2-i)2
Cancel the common factors.
Factor 2 out of 2.
2(2-i)2(1)
Cancel the common factor.
2(2-i)2⋅1
Rewrite the expression.
2-i1
Divide 2-i by 1.
2-i
2-i
2-i
Simplify (3+i)/(1+i)

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