Problem 27 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-34: Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ (2+5 i)^{3} $$

Step-by-Step Solution

Answer

(-142 - 65i)

Step 1: Expand the Expression

To write the complex number (2+5i)^3 in the form a + bi, we need to expand the expression. This is similar to using the binomial theorem for expansion. (2+5i)^3 = (2+5i)(2+5i)(2+5i).

Step 2: Calculate (2+5i)(2+5i)

First, find the square of the complex number: (2+5i)(2+5i) = 4 + 20i + 25i^2. Since i^2 = -1, we substitute this in to get: 4 + 20i - 25. Combine like terms: -21 + 20i.

Step 3: Multiply Result by (2+5i)

Now take the result from the previous step, (-21+20i), and multiply it by (2+5i): (-21+20i)(2+5i) = -42 - 105i + 40i + 100i^2.

Step 4: Simplify and Combine Like Terms

Combine the real and imaginary parts. Remember that i^2 = -1, so 100i^2 = -100. Thus, the expression becomes: -42 - 105i + 40i - 100. Combine the real parts: -42 - 100 = -142. Combine the imaginary parts: -105i + 40i = -65i.

Key Concepts

Binomial Theorem

The binomial theorem provides a systematic method of expanding expressions that are raised to a positive integer power, like \((a+b)^n\). It states that:

• Every term in the expansion is composed of a coefficient, a power of \(a\), and a power of \(b\).
• The general formula is given by the sum of terms: \(C(n, k) \cdot a^{n-k} \cdot b^k\), where \(C(n, k)\) are binomial coefficients.

In our problem, \((2+5i)^3\), we consider \(a=2\), \(b=5i\), and \(n=3\). By applying the binomial theorem, we'd expand this as:\[ (2+5i)^3 = C(3, 0) \cdot 2^3 \cdot (5i)^0 + C(3, 1) \cdot 2^2 \cdot (5i)^1 + C(3, 2) \cdot 2^1 \cdot (5i)^2 + C(3, 3) \cdot 2^0 \cdot (5i)^3 \]This method offers a conceptual view of breaking the expression into smaller, manageable parts, making it easier to compute.

Imaginary Unit

The imaginary unit, represented by \(i\), is a fundamental component of complex numbers. It is defined by the property:

• \(i^2 = -1\)

This definition allows us to deal with equations that have negative numbers under square roots, something which real numbers alone cannot accommodate. In the expression \((2+5i)^3\), the imaginary unit comes into play during multiplication:
1. When you square \((2+5i)\), you encounter \(25i^2\). Using the property of \(i\), it becomes \(-25\).2. Similarly, when further operations involve \(i\), we must constantly apply \(i^2 = -1\) to rewrite terms accordingly.The mastery of \(i\) is crucial; without it, handling complex number operations would be impossible.

Complex Number Multiplication

Multiplying complex numbers extends the distributive property familiar from real numbers to incorporate the imaginary unit. Follow the rule: for \[(a+bi)(c+di) = (ac-bd) + (ad+bc)i\]Here's how it applies specifically to our example:

• First, expand the multiplication: \((2+5i)(2+5i)\), which involves squaring 2, multiplying 2 with 5i, squaring 5i, and adding up all terms.
• Repeat the process for multiplying the resultant expression with \((2+5i)\).
• Keep in mind, each time you face \(i^2\), it converts to \(-1\), altering the term's sign from positive to negative.

Ultimately, what might initially seem complex becomes routine with practice. Breaking down operations into real and imaginary parts helps simplify the process, resulting in our final form \(-142 - 65i\).