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Chapter 3: Problem 16

Find the real and imaginary parts of $$ (3.00+i)^{3}+(6.00+5.00 i)^{2} $$

Short Answer

Expert verified
The expression simplifies to 30 + 53i.

Step by step solution

01

Expand the First Component

Start by expanding the first part of the equation, (3.00+i)^{3}. By using the binomial theorem, it can be expanded as \(3^3 + 3*3^2*i - 3*i^2 - i^3 = 19 - 7i.\)
02

Expand the Second Component

Next, expand the second part of the equation, (6.00+5.00 i)^{2}. Using the binomial theorem again, we get \(6^2 - 5^2 + 2*6*5* i = 11 + 60i.\)
03

Combine Real and Imaginary Parts

We add the real and imaginary parts from both the components together. The real parts: 19 (from Step 1) + 11 (from Step 2), and the imaginary parts: -7i (from Step 1) + 60i (from Step 2). The summation gives us 30 + 53i.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Real Part
In complex numbers, the real part is essentially the non-imaginary portion of the number. Complex numbers have the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.
For example, in the complex number \(3 + 4i\), 3 is the real part.
  • When adding or subtracting complex numbers, combine only their real parts. For instance, adding \(3 + 4i\) and \(5 + 6i\) would involve adding 3 and 5 to get the real part of the result: 8.
  • The real part can be found using the formula \(\text{Re}(a + bi) = a\).
In the given exercise, after expanding the expressions using the binomial theorem, the real parts are derived as follows:
The real component from \((3.00+i)^{3}\) is 19 and from \((6.00+5.00i)^{2}\) is 11. So when we add these, the total real part is 30.
Imaginary Part
The imaginary part in a complex number involves the coefficient of \(i\), denoted as \(b\) in the expression \(a + bi\).
Unlike real numbers, imaginary numbers are based on \(i\), where \(i = \sqrt{-1}\).
This enables us to work with equations that don't have real solutions due to negative square roots.
  • In calculations, like addition or subtraction, similar to the real part, the imaginary components are combined separately. For instance, in \(5 + 6i\) and \(7 + 8i\), we only work with 6 and 8 to get the new imaginary part, which is 14.
  • Imaginary parts are isolated in complex expressions using \(\text{Im}(a + bi) = b\).
For the exercise, the imaginary components are found separately after applying the binomial theorem:
From \((3.00+i)^{3}\), it is -7i, and from \((6.00+5.00i)^{2}\), it amounts to 60i. Summing them gives a total imaginary part of 53i.
Binomial Theorem
The binomial theorem provides a way to expand expressions that are raised to a power. It is especially useful in complex numbers for expressions like \((a + b)^n\).
  • This theorem states that \((a + b)^n\) can be expanded using the formula: \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\].
  • Each term is a product involving a binomial coefficient \(\binom{n}{k}\) and powers of \(a\) and \(b\).
In the original exercise:
1. The first expression \((3.00+i)^3\) expands into individual terms as \(3^3 + 3 \cdot 3^2 \cdot i - 3 \cdot i^2 - i^3\). The calculations yield the result 19 - 7i.
2. Similarly, the second \((6.00+5.00i)^2\) expands to \(6^2 - 5^2 + 2 \cdot 6 \cdot 5 \cdot i\), resulting in 11 + 60i.
The binomial theorem allows us to simplify and solve complex expressions systematically, breaking them into manageable real and imaginary components.

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