/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1016 Expand \((2+3 i)^{3}\)... [FREE SOLUTION] | 91影视

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Chapter 31: Problem 1016

Expand \((2+3 i)^{3}\)

Short Answer

Expert verified
\((2+3i)^3 = -46 + 9i\)

Step by step solution

01

Recall the binomial theorem

The binomial theorem states that for any positive integer n and complex numbers a and b, we have: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\) In this case, we have \(n=3\), \(a=2\), and \(b=3i\). We will now apply the binomial theorem to the given expression.
02

Expand the polynomial using the binomial theorem

Using the binomial theorem, we write the expression as a sum of binomial coefficients and powers of a and b: \((2+3i)^3 = \sum_{k=0}^{3} \binom{3}{k} (2)^{3-k} (3i)^{k}\) Now, we'll compute each term of the sum individually: - For \(k=0\): \[\binom{3}{0} (2)^{3-0} (3i)^{0} = 1 * 8 * 1 = 8\] - For \(k=1\): \[\binom{3}{1} (2)^{3-1} (3i)^{1} = 3 * 4 * (3i) = 36i\] - For \(k=2\): \[\binom{3}{2} (2)^{3-2} (3i)^{2} = 3 * 2 * (9i^2) = -54\] - For \(k=3\): \[\binom{3}{3} (2)^{3-3} (3i)^{3} = 1 * 1 * (27i^3) = -27i\]
03

Combine the terms

Now, we will combine these terms to get the final expression: \((2+3i)^3 = 8 + 36i - 54 - 27i = (-54+8) + (36-27)i\)
04

Simplify the expression

Finally, we will simplify the result: \((-46) + (9)i\) So, \((2+3i)^3 = -46 + 9i\).

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