Multiply -6i*(6-2i)

The question requires you to perform a multiplication operation between a pure imaginary number and a complex number. The pure imaginary number is $- 6 i$ , where $i$ is the imaginary unit with the property that $i^{2} = - 1$ . The complex number is written in binomial form as $6 - 2 i$ , which consists of a real part, $6$ , and an imaginary part, $- 2 i$ . The multiplication process involves using the distributive property to multiply $- 6 i$ by both terms in the complex number, and then combining like terms, taking into account the aforementioned property of the imaginary unit $i$ .

$- 6 i \cdot (6 - 2 i)$

Answer

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Solution:

Step:1

Implement the distributive property to expand the expression: $- 6 i \cdot (6 - 2 i)$ .

Step:2

First, multiply $6$ by $- 6 i$ : $- 6 i \cdot 6 - 6 i \cdot (- 2 i)$ .

Step:3

Proceed to multiply $- 6 i$ by $- 2 i$ .

Step:3.1

Calculate the product of $- 6$ and $- 2$ : $- 6 i \cdot 6 + (- 6 \cdot - 2) i^{2}$ .

Step:3.2

Consider $i$ raised to the first power: $- 6 i \cdot 6 + 12 i \cdot i$ .

Step:3.3

Again, consider $i$ raised to the first power: $- 6 i \cdot 6 + 12 i \cdot i$ .

Step:3.4

Apply the exponentiation rule $a^{m} \cdot a^{n} = a^{m + n}$ to simplify the expression: $- 6 i \cdot 6 + 12 i^{1 + 1}$ .

Step:3.5

Sum the exponents $1$ and $1$ : $- 6 i \cdot 6 + 12 i^{2}$ .

Step:4

Simplify the terms in the expression.

Step:4.1

Substitute $i^{2}$ with $- 1$ : $- 6 i \cdot 6 + 12 \cdot (- 1)$ .

Step:4.2

Multiply $12$ by $- 1$ : $- 6 i \cdot 6 - 12$ .

Step:5

Rearrange the terms to obtain the final result: $- 12 - 6 i \cdot 6$ .

Knowledge Notes:

The problem involves complex number multiplication. Here are the relevant knowledge points:

Complex Numbers: A complex number is of the form $a + b i$ where $a$ is the real part and $b i$ is the imaginary part. The imaginary unit $i$ is defined as $\sqrt{- 1}$ .
Distributive Property: This property states that $a (b + c) = a b + a c$ . It allows us to expand expressions where a single term is multiplied by a sum or difference.
Multiplication of Imaginary Numbers: When multiplying terms with the imaginary unit $i$ , we use the fact that $i^{2} = - 1$ . This helps in simplifying expressions involving powers of $i$ .
Exponentiation Rules: The power rule for exponents states that $a^{m} \cdot a^{n} = a^{m + n}$ . This is used when multiplying like bases with exponents.
Simplification: After applying the distributive property and multiplication rules, the expression is simplified by combining like terms and substituting powers of $i$ with their respective values.
Reordering Terms: The final step in simplifying expressions involving complex numbers often involves reordering the terms to present the result in standard form, $a + b i$ .