Problem 38
Question
Perform the indicated operation and write the result in standard form. $$-2 i(7+9 i)$$
Step-by-Step Solution
Verified Answer
Thus, the result of the given operation in standard form is \( 18 - 14i \).
1Step 1: Distribute the -2i to each term in the parenthesis
To simplify the expression, distribute real number -2i to the real part and the imaginary part of the parenthesis: \[ -2i \cdot 7 + (-2i \cdot 9i)\].
2Step 2: Perform the multiplication
Multiplication ties directly to the commutative property of multiplication, which states that the order in which you multiply numbers does not change the product. Hence, \[ -2i \cdot 7 = -14i\] and \[ -2i \cdot 9i = -18i^2\]. Since \(i^2 = -1\), this converts to +18.
3Step 3: Sum up the real and imaginary parts
After multiplication, the final expression will be the sum of the real part (18 from Step 2) and the imaginary part (-14i from Step 2), hence the result is \[ 18 - 14i \].
Key Concepts
Imaginary UnitMultiplication of Complex NumbersStandard Form of Complex Numbers
Imaginary Unit
In the world of complex numbers, the imaginary unit plays a crucial role. It is denoted by the symbol \(i\). The imaginary unit is unique because it is defined as the square root of negative one, hence, \(i^2 = -1\). This definition allows us to expand the concept of numbers beyond the real line and explore new dimensions of calculation.
Without the imaginary unit, we could not solve equations like \(x^2 + 1 = 0\), because there are no real numbers that satisfy this equation. But with \(i\), we find solutions \(x = i\) and \(x = -i\).
When performing operations with \(i\), like multiplication, it’s vital to remember that \(i^2\) must be substituted with \(-1\). This transforms expressions and is a key step in simplifying complex numbers
Without the imaginary unit, we could not solve equations like \(x^2 + 1 = 0\), because there are no real numbers that satisfy this equation. But with \(i\), we find solutions \(x = i\) and \(x = -i\).
When performing operations with \(i\), like multiplication, it’s vital to remember that \(i^2\) must be substituted with \(-1\). This transforms expressions and is a key step in simplifying complex numbers
Multiplication of Complex Numbers
Multiplying complex numbers involves applying the distributive property, just like in ordinary algebra. For example, to multiply \(-2i\) by \((7 + 9i)\), distribute \(-2i\) across each term inside the parentheses.
Start with \(-2i \times 7\), which equals \(-14i\). Then, calculate \(-2i \times 9i\). Here, you will first multiply the numbers in front, giving \(-18i^2\). Since \(i^2\) equals \(-1\), replace \(-18i^2\) with \(18\), converting the complex multiplication into a real number.
Thus, multiplication steps break down into:
Start with \(-2i \times 7\), which equals \(-14i\). Then, calculate \(-2i \times 9i\). Here, you will first multiply the numbers in front, giving \(-18i^2\). Since \(i^2\) equals \(-1\), replace \(-18i^2\) with \(18\), converting the complex multiplication into a real number.
Thus, multiplication steps break down into:
- Distribute each term
- Replace \(i^2\) with \(-1\)
- Simplify to get the final expression
Standard Form of Complex Numbers
A complex number is generally expressed in its standard form as \(a + bi\), where "\(a\)" is the real part, and "\(bi\)" is the imaginary part. The standard form is significant because it provides a clear structure for performing further mathematical operations.
In the exercise, after completing the multiplication, we obtained the expression \(18 - 14i\). Here, 18 is the real component, and \(-14i\) is the imaginary component.
Putting complex numbers in their standard form ensures they are ready for addition, subtraction, or even graphing on a coordinate plane. It's a critical step in concluding any operation involving complex numbers and ensures the result is clear and usable for further analysis or problem-solving. This form is essential for maintaining consistency in writing complex expressions.
In the exercise, after completing the multiplication, we obtained the expression \(18 - 14i\). Here, 18 is the real component, and \(-14i\) is the imaginary component.
Putting complex numbers in their standard form ensures they are ready for addition, subtraction, or even graphing on a coordinate plane. It's a critical step in concluding any operation involving complex numbers and ensures the result is clear and usable for further analysis or problem-solving. This form is essential for maintaining consistency in writing complex expressions.
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