Problem 38 - College Algebra with Modeling and Visualization

Question

Write the expression in standard form. $$ -i(5-2 i)^{2} $$

Step-by-Step Solution

Answer

The expression in standard form is \(-20 - 21i\).

Step 1: Expand the Square

We need to expand \((5 - 2i)^2\) using the formula \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = 5\) and \(b = 2i\). So, \((5 - 2i)^2 = 5^2 - 2(5)(2i) + (2i)^2\).

Step 2: Calculate Each Term

Calculate \(5^2 = 25\), \(-2(5)(2i) = -20i\), and \((2i)^2 = 4i^2\). Remember that \(i^2 = -1\), so \(4i^2 = 4(-1) = -4\).

Step 3: Combine the Terms

Combine the results from Step 2: \(25 - 20i - 4\). Simplify by combining the real parts: \(25 - 4 = 21\). Therefore, \((5 - 2i)^2 = 21 - 20i\).

Step 4: Multiply by -i

Now, multiply the result \((21 - 20i)\) by \(-i\): \(-i(21 - 20i) = -i \cdot 21 + (-i) \cdot (-20i)\).

Step 5: Distribute and Simplify

Calculate \(-i \cdot 21 = -21i\) and \(-i \cdot (-20i) = 20i^2\). Since \(i^2 = -1\), we have \(20(-1) = -20\).

Step 6: Combine Real and Imaginary Parts

Combine the terms to form the final expression: The real part is \(-20\) and the imaginary part is \(-21i\). Therefore, the expression in standard form is \(-20 - 21i\).

Key Concepts

Imaginary Unit

The imaginary unit, represented by the letter \(i\), is a fundamental concept in complex numbers and differs from other numbers because it is defined as the square root of \(-1\). This definition leads to interesting properties: primarily, it helps us express the square root of negative numbers as products involving \(i\).

Some essential rules and properties of the imaginary unit \(i\) include:

• \(i^2 = -1\): This is the core property that supports its use in calculations involving imaginary and complex numbers.
• \(i^3 = -i\): Derived from \(i^2\), this helps simplify higher powers of \(i\).
• \(i^4 = 1\): After this, the powers of \(i\) repeat in cycles of four: \(i^5 = i\),\(i^6 = -1\), etc.

When calculating expressions involving complex numbers, we need the properties of \(i\) to simplify terms, such as transforming \(i^2\) effectively into \(-1\), as seen in the exercise’s calculation of \(4i^2\) into \(-4\). Understanding these properties allows us to solve complex number operations confidently.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations like addition, subtraction, multiplication, and division. In the context of complex numbers, they often include \(i\) as a variable-like symbol representing the imaginary unit.

Expanding and simplifying algebraic expressions containing complex numbers follows standard algebraic rules, but with extra care taken for the imaginary unit:\

• When expanding squares, such as \((5 - 2i)^2\), apply the formula \((a - b)^2 = a^2 - 2ab + b^2\).
• Be cautious with terms containing \(i\) to properly incorporate \(i^2 = -1\), which affects the sign of resulting components.

In the exercise provided, the expansion resulted in terms like \(25 - 20i + 4(-1)\), which simplify to real parts \(21\) and imaginary parts \(-20i\). This reapplies the core notion that any surface-level operation in algebra is deeply affected by the properties of \(i\) when working with complex numbers.

Mathematical Notation

Mathematical notation provides a crucial framework for expressing complex ideas simply and universally. In complex numbers, it's necessary for describing operations involving both real components and the otherwise non-existent imaginary terms.

Standard notation for complex numbers uses the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. This format allows for clear expression and manipulation of numbers that aren’t easily grouped with real numbers alone:

• The real component \(a\) behaves like regular real numbers in calculations.
• The imaginary component \(bi\) includes the variable \(i\), with its unique properties affecting how we treat the expression.

For mathematical clarity and consistency, the exercise results in standard form \(-20 - 21i\). It labels \(-20\) as the real part and \(-21i\) as the imaginary part, appreciating that adopting such notation helps in understanding and solving further complex equations.