Write each complex number in standard form. (2 + 3i)³

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1. Rewrite the expression \((2 + 3i)^3\) using the binomial theorem: \((a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\), where \(a = 2\), \(b = 3i\), and \(n = 3\).
2. Expand the binomial using the formula: \((2 + 3i)^3 = \binom{3}{0}(2)^3(3i)^0 + \binom{3}{1}(2)^2(3i)^1 + \binom{3}{2}(2)^1(3i)^2 + \binom{3}{3}(2)^0(3i)^3\).
3. Simplify each term in the expansion: \(\binom{3}{0}(2)^3(3i)^0 = 8\), \(\binom{3}{1}(2)^2(3i)^1 = 36i\), \(\binom{3}{2}(2)^1(3i)^2 = -54\), and \(\binom{3}{3}(2)^0(3i)^3 = -27i\).
4. Combine the real and imaginary parts: Add the real terms \(8 - 54\) and the imaginary terms \(36i - 27i\).
5. Write the result in standard form \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of \(i\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined as the square root of -1. Understanding complex numbers is essential for performing operations such as addition, subtraction, multiplication, and division, as well as for converting them into standard form.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a and b are real numbers. To express a complex number in standard form, one must ensure that the imaginary unit i is isolated in the second term. This is particularly important when performing operations like exponentiation, as the result must be simplified to fit this format.

Exponentiation of Complex Numbers

Exponentiation of complex numbers involves raising a complex number to a power, which can be done using the binomial theorem or by expanding the expression. In the case of (2 + 3i)^3, one must apply the distributive property or binomial expansion to calculate the result accurately. Understanding how to manipulate and simplify these expressions is crucial for arriving at the correct standard form.