4+5i1-3i

Multiply the numerator and denominator of 4+5i1-3i by the conjugate of 1-3i to make the denominator real.

4+5i1-3i⋅1+3i1+3i

Combine.

(4+5i)(1+3i)(1-3i)(1+3i)

Simplify the numerator.

Expand (4+5i)(1+3i) using the FOIL Method.

Apply the distributive property.

4(1+3i)+5i(1+3i)(1-3i)(1+3i)

Apply the distributive property.

4⋅1+4(3i)+5i(1+3i)(1-3i)(1+3i)

Apply the distributive property.

4⋅1+4(3i)+5i⋅1+5i(3i)(1-3i)(1+3i)

4⋅1+4(3i)+5i⋅1+5i(3i)(1-3i)(1+3i)

Simplify and combine like terms.

Simplify each term.

Multiply 4 by 1.

4+4(3i)+5i⋅1+5i(3i)(1-3i)(1+3i)

Multiply 3 by 4.

4+12i+5i⋅1+5i(3i)(1-3i)(1+3i)

Multiply 5 by 1.

4+12i+5i+5i(3i)(1-3i)(1+3i)

Multiply 5i(3i).

Multiply 3 by 5.

4+12i+5i+15ii(1-3i)(1+3i)

Raise i to the power of 1.

4+12i+5i+15(i1i)(1-3i)(1+3i)

Raise i to the power of 1.

4+12i+5i+15(i1i1)(1-3i)(1+3i)

Use the power rule aman=am+n to combine exponents.

4+12i+5i+15i1+1(1-3i)(1+3i)

Add 1 and 1.

4+12i+5i+15i2(1-3i)(1+3i)

4+12i+5i+15i2(1-3i)(1+3i)

Rewrite i2 as -1.

4+12i+5i+15⋅-1(1-3i)(1+3i)

Multiply 15 by -1.

4+12i+5i-15(1-3i)(1+3i)

4+12i+5i-15(1-3i)(1+3i)

Subtract 15 from 4.

-11+12i+5i(1-3i)(1+3i)

Add 12i and 5i.

-11+17i(1-3i)(1+3i)

-11+17i(1-3i)(1+3i)

-11+17i(1-3i)(1+3i)

Simplify the denominator.

Expand (1-3i)(1+3i) using the FOIL Method.

Apply the distributive property.

-11+17i1(1+3i)-3i(1+3i)

Apply the distributive property.

-11+17i1⋅1+1(3i)-3i(1+3i)

Apply the distributive property.

-11+17i1⋅1+1(3i)-3i⋅1-3i(3i)

-11+17i1⋅1+1(3i)-3i⋅1-3i(3i)

Simplify.

Multiply 1 by 1.

-11+17i1+1(3i)-3i⋅1-3i(3i)

Multiply 3 by 1.

-11+17i1+3i-3i⋅1-3i(3i)

Multiply -3 by 1.

-11+17i1+3i-3i-3i(3i)

Multiply 3 by -3.

-11+17i1+3i-3i-9ii

Raise i to the power of 1.

-11+17i1+3i-3i-9(i1i)

Raise i to the power of 1.

-11+17i1+3i-3i-9(i1i1)

Use the power rule aman=am+n to combine exponents.

-11+17i1+3i-3i-9i1+1

Add 1 and 1.

-11+17i1+3i-3i-9i2

Subtract 3i from 3i.

-11+17i1+0-9i2

Add 1 and 0.

-11+17i1-9i2

-11+17i1-9i2

Simplify each term.

Rewrite i2 as -1.

-11+17i1-9⋅-1

Multiply -9 by -1.

-11+17i1+9

-11+17i1+9

Add 1 and 9.

-11+17i10

-11+17i10

-11+17i10

Split the fraction -11+17i10 into two fractions.

-1110+17i10

Move the negative in front of the fraction.

-1110+17i10

Write in Standard Form (4+5i)/(1-3i)