Question

Multiply. Write the answer in standard form. \((9+5 i)(9-5 i)\)

Short answer

The multiplication of \((9+5i)(9-5i)\) in standard form is \(106\).

Step-by-step solution

Distribute

Like the FOIL method in algebra, multiply the 'First' terms, the 'Outer' terms, the 'Inner' terms and the 'Last' terms; so we get: \(9*9, 9*-5i, 5i*9, 5i*-5i\)

Simplify

By multiplying the values together, the equation becomes \(81 - 45i + 45i - 25i^2\). Note, \(i^2 = -1\)

Replace \(i^2\)

Substitute -1 for \(i^2\), the equation becomes \(81 - 45i + 45i + 25\)

Combine like terms

The imaginary parts cancel each other out and adding the real parts we get \(106\).

Key concepts

Standard Form in Algebra

The standard form in algebra is a way of writing numbers or equations so that they follow a specific format. For complex numbers, which include both real and imaginary components, the standard form is usually written as \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part, with \( i \) representing the square root of -1.

When complex numbers are multiplied and need to be expressed in standard form, the result should be simplified such that no imaginary numbers are multiplied by each other. This usually involves combining like terms and simplifying the constants. In the given exercise \((9+5i)(9-5i)\), the final answer in standard form is a single number, 106, which is purely real, indicating that the imaginary components have cancelled each other out through the multiplication process.

FOIL Method

FOIL is a mnemonic for First, Outer, Inner, Last and is a technique used to multiply two binomials. It ensures that each term in the first binomial is multiplied by each term in the second binomial. Here's how the FOIL method works when applied to complex numbers:

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outer terms.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms.

For the exercise, using FOIL we multiply \(9\) by \(9\) (first), \(9\) by \(-5i\) (outer), \(5i\) by \(9\) (inner), and \(5i\) by \(-5i\) (last). This methodical approach ensures all parts of the binomials are accounted for in the multiplication.

Combining Like Terms

Combining like terms is a fundamental process in algebra used to simplify expressions. Like terms are terms that have the same variables raised to the same powers. In the context of complex numbers, combining like terms involves adding or subtracting the real parts and the imaginary parts separately. After applying the FOIL method, as seen in the example \(81 - 45i + 45i - 25i^2\), you can notice that the terms \(-45i\) and \(+45i\) are like terms and cancel each other out because they are equal and opposite. This leaves us with just the real parts, 81 and 25, which can be combined to reach the final solution.

Imaginary Numbers

Imaginary numbers are an extension of the real number system. They are necessary to solve equations where a number must be squared to get a negative result, which is not possible with only real numbers. The basic unit of imaginary numbers is \( i \), which is defined as \( \sqrt{-1} \). Any time you square \( i \), you get -1.

In the context of our exercise, when we multiply the last terms, \(5i * -5i\), we get \(-25i^2\). Since \(i^2 = -1\), this product simplifies to \(25\), a real number. Understanding how imaginary numbers work and interact with real numbers is critical in simplifying complex number expressions and maintaining them in standard form.