Multiplying Complex Numbers

Multiplying complex numbers is very similar to multiplying in the real number system. The exponent rules will be used as well as the distributive property. There will be one additional step where the imaginary components with exponents will be replaced with its simplest form.

Quick Review:

i	$\sqrt{- 1}$
i With exponents	Simplest form
i²	-1
i³	-i
i⁴	1

Multiplying monomials

(3i)(-4i) (-4i²)(5i) Exponent Rule

= -12 i² = -20 i³

x^{a} • x^{b} = x^{a + b}

= -12 (-1)= -20 (-i)

= 12 = 20i

Multiply a polynomial by a monomial

4i(2i² - 3i + 7)

= 4i(2i² - 3i + 7) Use the distributive property

= 8i³ - 12i² + 28i

= 8(-i) - 12 (-1) + 28i replace the imaginary numbers with exponents to the simplest form

= -8i + 12 + 28i simplify

= 12 + 20i combine like terms and write in a + bi form

Multiply a binomial by a binomial

(5+2i)(2 - 6i)

= 10 - 30i + 4i - 12i²FOIL or distributive property

= 10 - 26i - 12i²   combine like terms

= 10 - 26i - 12(-1)   replace the imaginary numbers with exponents to the   simplest form

= 22 -26i      combine like terms and write in a + bi form

(2 - 3i²)(2 + 3i²)

= 4 + 6i² - 6i² -9i⁴ FOIL or distributive property

= 4 - 9i⁴      combine like terms

= 4 - 9(1)    replace the imaginary numbers with exponents to the simplest form

= -5   combine like terms

Multiply Complex Numbers by using the rules for the real numbers then replace the imaginary number with exponents with its simplest form and simplify. When applicable make sure that the answer is in a + bi form.

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Multiplying Complex Numbers

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