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 $-6i$, where $i$is the imaginary unit with the property that $i^2 = -1$. The complex number is written in binomial form as $6 - 2i$, which consists of a real part, $6$, and an imaginary part, $-2i$. The multiplication process involves using the distributive property to multiply $-6i$by both terms in the complex number, and then combining like terms, taking into account the aforementioned property of the imaginary unit $i$.
$- 6 i \cdot \left(\right. 6 - 2 i \left.\right)$
Answer
Solution:
Step:1
Implement the distributive property to expand the expression: $-6i \cdot (6 - 2i)$.
Step:2
First, multiply $6$ by $-6i$: $-6i \cdot 6 - 6i \cdot (-2i)$.
Step:3
Proceed to multiply $-6i$ by $-2i$.
Step:3.1
Calculate the product of $-6$ and $-2$: $-6i \cdot 6 + (-6 \cdot -2)i^2$.
Step:3.2
Consider $i$ raised to the first power: $-6i \cdot 6 + 12i \cdot i$.
Step:3.3
Again, consider $i$ raised to the first power: $-6i \cdot 6 + 12i \cdot i$.
Step:3.4
Apply the exponentiation rule $a^m \cdot a^n = a^{m+n}$ to simplify the expression: $-6i \cdot 6 + 12i^{1+1}$.
Step:3.5
Sum the exponents $1$ and $1$: $-6i \cdot 6 + 12i^2$.
Step:4
Simplify the terms in the expression.
Step:4.1
Substitute $i^2$ with $-1$: $-6i \cdot 6 + 12 \cdot (-1)$.
Step:4.2
Multiply $12$ by $-1$: $-6i \cdot 6 - 12$.
Step:5
Rearrange the terms to obtain the final result: $-12 - 6i \cdot 6$.
Knowledge Notes:
The problem involves complex number multiplication. Here are the relevant knowledge points:
Complex Numbers: A complex number is of the form $a + bi$ where $a$ is the real part and $bi$ is the imaginary part. The imaginary unit $i$ is defined as $\sqrt{-1}$.
Distributive Property: This property states that $a(b + c) = ab + ac$. 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, $a + bi$.
