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Chapter 7: Problem 64

Multiply. Give answers in standard form. $$3 i(-3-i)^{2}$$

Short Answer

Expert verified
-18 + 24i

Step by step solution

01

- Expand the expression

First, expand the expression \((-3 - i)^{2}\). Use the formula for the square of a binomial: \(a^2 + 2ab + b^2\). Here, \(a = -3\) and \(b = -i\).
02

- Calculate each term of the expansion

Calculate the square for each term: \[(-3)^{2} = 9\] \[2(-3)(-i) = 6i\] \[(-i)^{2} = -1\] Combine these: \[9 + 6i - 1 = 8 + 6i\]
03

- Distribute and simplify

Now distribute \(3i\) over the expanded expression: \[3i(8 + 6i)\] Use the distributive property: \(3i \times 8 + 3i \times 6i\)Calculate each part: \[24i\] \[(3i)(6i) = 18i^2\] Recall that \(i^2 = -1\), so \(18i^2 = 18(-1) = -18\)Combine the results: \[24i - 18\]
04

- Write the final answer in standard form

In standard form, combine the real and imaginary parts: \(-18 + 24i\)

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Key Concepts

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

Binomial Expansion
Binomial expansion is a technique used to expand expressions raised to a power. In our example, we needed to expand \((-3 - i)^{2}\). A binomial is an expression with two terms. Here, \(a = -3\) and \(b = -i\). The square of a binomial formula is \(a^2 + 2ab + b^2\). Start by squaring each term: \[ (-3)^2 = 9 \] and \[ (-i)^2 = -1 \]. Then, calculate the middle term: \[2(-3)(-i) = 6i\]. Combining these gives us \[9 + 6i - 1 = 8 + 6i\]. Binomial expansion makes it easier to handle expressions involving powers, providing a straightforward approach to break down complex terms.
Complex Numbers
Complex numbers have a real part and an imaginary part, represented as \(a + bi\). In our problem, we dealt with \(3i\), which is an imaginary number (real part = 0) and \(-3 - i\), which includes both real (\(-3)\) and imaginary (\(-i)\) parts. Complex numbers are crucial in many fields, such as engineering and physics. They follow unique rules: \(i^2 = -1\). This specific rule was used to simplify \(18i^2\) to \(-18\). Handling complex numbers requires understanding these individual rules and combining them logically to simplify the expression.
Distributive Property
The distributive property helps us multiply a single term by each term in a binomial. In the problem, we distributed \(3i\) over the expanded binomial \(8 + 6i\), resulting in: \[3i(8 + 6i)\]. Applying the distributive property, we get \[3i \times 8 + 3i \times 6i\]. Calculating each part: \[24i\] and \[(3i)(6i) = 18i^2 = -18\]. Combining results: \[24i - 18\]. This property is essential for breaking down expressions into manageable parts, ensuring correct calculations when dealing with multiple terms.

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