Question

Write the expression as a complex number in standard form. \((3+5 i)\left(2-7 i^4\right)\)

Short answer

The complex number in standard form is \(-15 - 25i\).

Step-by-step solution

Recognize the simplification

First, notice that \(i^4 = 1\) because \(i\) is the imaginary unit and by definition \(i^2 = -1\), hence \(i^4 = (i^2)^2 = 1\). Use this to simplify the term inside the bracket to \(2 - 7\).

Simplify the brackets

After simplification, the expression becomes \((3+5i)(2-7)\)

Distribute the multiplication

To multiply, distribute each term in the first bracket with each term in the second bracket, to get \((3*2 + 3*-7 + 5*i*2 + 5*i*-7)\)

Simplify the multiplication

When you perform the multiplication, you get: \((6 - 21 + 10i - 35i)\)

Combine like terms

Finally, combine the real parts and the imaginary parts into the final answer: \(-15 - 25i\)

Key concepts

Imaginary Numbers

Imaginary numbers are numbers that involve the square root of a negative number. They are key elements in complex numbers, which combine real and imaginary numbers. The fundamental imaginary unit "i" is defined as the square root of -1. Therefore, performing operations with 'i', such as powering it, will lead to different outcomes.

• When you multiply 'i' by itself once, we have: \(i^2 = -1\).
• Multiplying 'i' four times gives us: \(i^4 = (i^2)^2 = 1\). This concept returns a real number and plays a vital role in simplifying expressions involving powers of 'i'.
  
• Imaginary numbers are used in engineering and physics, often found in wave functions and electrical engineering calculations.

Standard Form

The standard form of a complex number is an orderly way to express it as a mixture of a real number and an imaginary number. It looks like this: \(a + bi\),
where:

• \(a\) is the real part of the complex number,
• \(b\) is the imaginary part, and
• "i" stands for the imaginary unit.

Having a standard form helps with operations like addition, subtraction, and multiplication of complex numbers. It allows you to clearly differentiate between the real and imaginary components, which is essential for simplifying expressions and solving equations.
By expressing complex numbers in standard form, it's easier to see their contribution in mathematical solutions and different fields that utilize complex analysis.

Multiplication of Complex Numbers

Multiplying complex numbers involves applying the distributive property, similar to multiplying binomial expressions in algebra.
To multiply two complex numbers, such as \((a + bi)(c + di)\), follow these steps:

• Multiply each term in the first complex number by each term in the second complex number.
• Combine all the resulting products: \(a * c + a * di + bi * c + bi * di\).
• Recognize that \(i^2 = -1\), which simplifies terms involving \(bi * di\) into real numbers.
• Group and simplify like terms: real terms together, and imaginary terms together.

This approach is similar to the FOIL method used in binomial multiplication. Once you've multiplied and combined like terms, your result will typically be a complex number in standard form.

Algebra 2

Algebra 2 extends the concepts learned in Algebra 1 and introduces students to more advanced topics, including complex numbers.
Understanding complex numbers is crucial for students as they progress through higher levels of math and it includes:

• Working with expressions containing 'i', the imaginary unit.
• Solving quadratic equations with no real solutions by using complex numbers.
• Exploring the properties of real and imaginary numbers and their role in polynomials.

Through this learning, students appreciate the versatility arithmetic operations have when applied to complex numbers. Especially, how complex numbers bridge between algebraic and complex analytical methods, offering solutions in applied mathematics, engineering, and physics. Algebra 2 provides the foundational knowledge to handle complex numbers fluently, set for future challenges like calculus and beyond.