veceriraby

Answered

2022-02-02

How do you write this expression in the standard form $a+bi$ given ${\left(1-i\right)}^{5}$?

Answer & Explanation

oferenteoo

Expert

2022-02-03Added 12 answers

Step 1
Since any exponent on the first term of 1 is simply 1, we can ignore that term.
Pay attention to the second term -i and the exponents on that term from the Binomial expansion plus use the Pascal triangle coefficients: 1, 2, 10, 10, 5, 1
${\left(1-i\right)}^{5}=\left(1\right){\left(-i\right)}^{5}+\left(5\right){\left(-i\right)}^{4}+\left(10\right){\left(-i\right)}^{3}+\left(10\right){\left(-i\right)}^{2}+\left(5\right){\left(-i\right)}^{1}+\left(1\right){\left(-i\right)}^{0}$
$=1\left(-i\right)+5\left(1\right)+10\left(i\right)+10\left(-1\right)+5\left(-i\right)+1\left(1\right)$
$=-4+4i$

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