Problem 100 - Algebra and Trigonometry

Question

Exercises \(99-101\) will help you prepare for the material covered in the next section. Refer to Section 1.4 if you need to review the basics of complex numbers. In each exercise, perform the indicated operation and write the result in the standard form \(a+b i .\) $$(-1+i \sqrt{3})(-1+i \sqrt{3})(-1+i \sqrt{3})$$

Step-by-Step Solution

Answer

The result of the given complex number multiplication is \(-1 + 6i \sqrt{3}\).

Step 1: Identify complex numbers

Identify the number as complex number; in this case, they are \(-1+i \sqrt{3}\), \(-1+i \sqrt{3}\) and \(-1+i \sqrt{3}\). They are of form \(a+ bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

Step 2: Multiply the first two numbers

Using the method of multiplication of binomials, multiply the first two numbers: \((-1+i \sqrt{3}) × (-1+i \sqrt{3})\). Like binomials, multiply real parts with real, real with imaginary and imaginary part with imaginary, collect like terms and simplify.

Step 3: Simplify the result

After multiplication, the result is \(1 - 2i\sqrt{3} - 3i^2\). Replace \(i^2\) with \(-1\) (since \(i = \sqrt{-1}\)) to get \(1 - 2i\sqrt{3} + 3\), or simplifying this we get \(4 - 2i\sqrt{3}\).

Step 4: Multiply with Third number

Now multiply this result with the third number \(-1 + i \sqrt{3}\): \[(4 - 2i\sqrt{3})(-1 + i\sqrt{3})\]. Again use multiplication of binomials.

Step 5: Final Simplification

After multiplication, the result is \(-4 + 2\sqrt{3}i + 4i \sqrt{3} - 3i^2\). Replace \(i^2\) with \(-1\) (since \(i = \sqrt{-1}\)) to get \(-4 + 2\sqrt{3}i + 4i\sqrt{3} + 3\), or simplifying this we get \(-1 + 6i \sqrt{3}\).

Key Concepts

Complex Numbers Standard Form

Understanding the standard form of complex numbers is fundamental when dealing with tasks involving these unique figures. A complex number is expressed in the form of \(a + bi\), where \(a\) refers to the real part and \(bi\) signifies the imaginary part. Here, \(i\) represents the imaginary unit, which is the square root of \(-1\). The beauty of the standard form is its simplicity in representing two-dimensional numbers on the complex plane, allowing for straightforward visualization and manipulation.

For instance, the complex number \(-1+i\sqrt{3}\), present in the given exercise, is in standard form with real part \(-1\) and imaginary part \(i\sqrt{3}\). Keeping complex numbers in this form is crucial, as it prepares us to apply operations like addition, subtraction, and, as in our exercise, multiplication, with clarity and ease.

Binomial Multiplication

In our exercise, binomial multiplication is the tool we employ to combine complex numbers. This process is similar to multiplying two binomial expressions in algebra, where we apply the distributive property, often known as the FOIL method (First, Outer, Inner, Last). This technique requires us to multiply each term in the first binomial with every term in the second.

For the initial multiplication in the provided exercise, one must take care to multiply \((-1+i\sqrt{3})\) by itself. We distribute as follows: \((-1) \times (-1)\), \((-1) \times (i\sqrt{3})\), \((i\sqrt{3}) \times (-1)\), and finally \((i\sqrt{3}) \times (i\sqrt{3})\). This process combines real and imaginary products to yield a new complex number, which is then further simplified by utilizing the properties of the imaginary unit.

Imaginary Unit Properties

Diving deeper into the multiplication steps, we encounter the properties of the imaginary unit \(i\). It is essential to remember that \(i^2 = -1\). This relationship is a critical component when simplifying the products of complex numbers.

After performing the initial round of binomial multiplication, we are left with terms that include \(i^2\). As showcased in the exercise, after multiplying the first two numbers, we get a term \(-3i^2\), which simplifies to \(3\) since \(i^2\) is replaced with \(-1\). This substitution is key in reducing the expression to its standard form, a process repeated each time we encounter a power of \(i\) in computation.

The comprehensive understanding of these properties not only streamlines the simplification process but also helps prevent common errors when performing operations with complex numbers, ensuring accuracy in the final result.