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The distributive property is a fundamental principle in mathematics. It's used to simplify and solve equations. The property states that multiplying a single term by a group of terms inside a parentheses is the same as doing the multiplication for every term separately.

Here's how it works: If you have an expression like \( a(b + c) \), you distribute \( a \) across each of the terms in the parenthesis:

• Multiply \( a \) by \( b \): \( ab \)
• Multiply \( a \) by \( c \): \( ac \)

Putting it all together, you get: \( ab + ac \).

For complex numbers, the distributive property helps in multiplying two binomials, such as \((2+15i)\) and \((2-15i)\). Each term from the first binomial multiplies with each term from the second, expanding the expression into four products. This step is crucial in solving complex multiplication and ensuring each part of the equation is addressed separately.

Standard form for complex numbers is a way to write them in a consistent and clear manner. A complex number is usually represented as \(a + bi\), where:

• \( a \) is the real part.
• \( b \) is the imaginary part.
• \( i \) is the imaginary unit, equal to \( \sqrt{-1} \).

Writing complex numbers in standard form allows for easy comparison and arithmetic operations. To place an expression in standard form, you need to separate the real and imaginary components and make sure they are clearly defined.

In the given exercise, after applying the distributive property and simplifying, we end up with \( 229 \), which is in the standard form since there is no imaginary part. This shows that operations involving complex numbers can have real number results, illustrating the versatility and utility of complex numbers in mathematics.

Imaginary numbers extend our understanding of numbers beyond the real line. The foundational element of imaginary numbers is \( i \), the imaginary unit, which is defined by \( i^2 = -1 \).

One of the key rules when working with imaginary numbers is recognizing that squaring the imaginary unit results in a real number. In calculations:

• \( i^2 \) should be replaced with \( -1 \).
• This transformation simplifies complex expressions by eliminating the non-real unit.

In the example problem, this rule was used when simplifying the product \(-225i^2\). By substituting \(-1\) for \(i^2\), the expression becomes \(225\), a real number.

These systematic steps ensure clarity and prevent errors when handling calculations involving complex numbers. Mastery of these rules is crucial for anyone working with higher mathematics that involves complex variables.