Question

Multiply. Write the answer in standard form. \((7+5 i)(8-6 i)\)

Short answer

The product of \((7+5 i)(8-6 i)\) in standard form is \(86-2i\).

Step-by-step solution

Distribute

Distribute across the parentheses like multiplying in regular algebra. So we have \((7*8)+(7*(-6i))+(5i*8)+(5i*(-6i))\).

Multiply

Perform the multiplication. So we get \(56 - 42i + 40i - 30i^2\).

Simplify

Simplify the expression, keep in mind that \(i^2 = -1\). So, \(56 - 42i + 40i + 30=86-2i\).

Key concepts

Standard Form of Complex Numbers

The standard form of a complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) represents the imaginary unit, satisfying the equation \(i^2 = -1\). In this format, \(a\) is known as the real part and \(b\) is the imaginary part of the complex number.

For example, if we have the complex number \(7+5i\), 7 is the real part and 5 is the imaginary part. It's crucial in mathematics to express complex numbers in this standard form as it simplifies the process of performing operations such as addition, subtraction, multiplication, and even division.

When we multiply complex numbers, the result should also be expressed in this standard form. To achieve this, we use algebraic rules including the distributive property and combine like terms, ensuring that the final expression is neat and appropriately ordered as a sum of a real number and an imaginary number.

Distributive Property

The distributive property is a cornerstone in algebra, playing a pivotal role in simplifying expressions and solving equations. In the context of complex numbers, it allows us to multiply a complex number by another by distributing each term of the first complex number to every term of the second.

Here's how it works in practice: Given two complex numbers in standard form, say \((a+bi)(c+di)\), we apply the distributive property to multiply each term of the first complex number (\(a+bi\)) by each term of the second (\(c+di\)).

• Multiplying \(a\) by both \(c\) and \(di\)
• Multiplying \(bi\) by both \(c\) and \(di\)

By systematically working through these multiplications, we ensure that every possible product is accounted for before collecting like terms to simplify to the standard form.

Imaginary Unit Arithmetic

Imaginary unit arithmetic revolves around the special number \(i\), defined by the property that \(i^2 = -1\). It is an essential concept in multiplying and handling complex numbers. Whenever you encounter the multiplication of two imaginary units, you must remember that it resolves to a real number because of this fundamental property.

Let's break down the arithmetic involving \(i\) with an example. In the multiplication \((5i)(-6i)\), each \(i\) represents the square root of -1, and when we multiply them together, we get \((-6)(5)(i^2) = -30i^2\). Recalling that \(i^2 = -1\), our expression simplifies to \(-30(-1)\), or \(30\), a real number!

This understanding is pivotal when multiplying complex numbers and simplifying expressions to their standard form. It allows us to convert terms with \(i^2\) back into real numbers, maintaining the integrity of the complex number's standard form.