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Math  /  Algebra

QuestionSimplify the expression. (3−5i)(8−i)(3-5 i)(8-i)

Studdy Solution

STEP 1

1. We are given a complex number expression to simplify.
2. The expression involves multiplication of two complex numbers.
3. Simplification involves expanding the product and combining like terms.

STEP 2

1. Expand the product using the distributive property.
2. Combine like terms to simplify the expression.

STEP 3

Use the distributive property (also known as the FOIL method for binomials) to expand the expression (3−5i)(8−i)(3-5i)(8-i).
(3−5i)(8−i)=3×8+3×(−i)+(−5i)×8+(−5i)×(−i) (3-5i)(8-i) = 3 \times 8 + 3 \times (-i) + (-5i) \times 8 + (-5i) \times (-i)

STEP 4

Calculate each term from the expansion:
1. 3×8=243 \times 8 = 24
2. 3×(−i)=−3i3 \times (-i) = -3i
3. (−5i)×8=−40i(-5i) \times 8 = -40i
4. (−5i)×(−i)=5i2(-5i) \times (-i) = 5i^2
Since i2=−1i^2 = -1, substitute −1-1 for i2i^2:
5i2=5(−1)=−5 5i^2 = 5(-1) = -5

STEP 5

Combine the calculated terms:
24−3i−40i−5 24 - 3i - 40i - 5
Combine like terms:
1. Real parts: 24−5=1924 - 5 = 19
2. Imaginary parts: −3i−40i=−43i-3i - 40i = -43i
Thus, the simplified expression is:
19−43i 19 - 43i

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