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Math  /  Algebra

QuestionFind the product and write the result in standard form. (3−8i)(−7−2i)(3-8 i)(-7-2 i)

Studdy Solution

STEP 1

1. The expression (3−8i)(−7−2i)(3 - 8i)(-7 - 2i) involves complex numbers.
2. We will use the distributive property (also known as the FOIL method for binomials) to find the product.
3. The result will be written in standard form, a+bia + bi, where aa and bb are real numbers.

STEP 2

1. Apply the distributive property to expand the product.
2. Simplify the expression by combining like terms.
3. Write the result in standard form.

STEP 3

Apply the distributive property (FOIL method) to expand the product:
(3−8i)(−7−2i)=3(−7)+3(−2i)+(−8i)(−7)+(−8i)(−2i)(3 - 8i)(-7 - 2i) = 3(-7) + 3(-2i) + (-8i)(-7) + (-8i)(-2i)
Calculate each term:
- First: 3×−7=−213 \times -7 = -21 - Outer: 3×−2i=−6i3 \times -2i = -6i - Inner: −8i×−7=56i-8i \times -7 = 56i - Last: −8i×−2i=16i2-8i \times -2i = 16i^2

STEP 4

Simplify the expression by combining like terms and using the fact that i2=−1i^2 = -1:
−21−6i+56i+16i2-21 - 6i + 56i + 16i^2
Combine the imaginary terms:
−21+50i+16(−1)-21 + 50i + 16(-1)
Simplify further:
−21+50i−16-21 + 50i - 16
Combine the real terms:
−37+50i-37 + 50i

STEP 5

Write the result in standard form a+bia + bi:
The product is:
−37+50i\boxed{-37 + 50i}

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