Problem 92 - Precalculus with Limits

Question

Finding the \(n\) th Roots of a Complex Number \(\operatorname{In}\) Exercises \(81-96,(\) a) use the formula on page 446 to find the indicated roots of the complex number, (b) represent each of the roots graphically, and (c) write each of the roots in standard form. Cube roots of $$1000$$

Step-by-Step Solution

Answer

The cube roots of 1000 are \(10, -5 + 5\sqrt{3}i, -5 - 5\sqrt{3}i\).

Step 1: Calculate the Cube Roots

The cube roots of a number \(n\) are given by \(n^{1/3}\). Therefore, the cube roots of 1000 can be calculated as \(1000^{1/3}\).

Step 2: Find the Roots

The cube root of 1000 is 10. Therefore, the cube roots of 1000 are 10, -5 + 5\(\sqrt{3}i\) and -5 - 5\(\sqrt{3}i\). We use the property of cube roots here, which states that if \(a\) is a cube root of \(n\), then there exist two other numbers, \(-a/2 + (\sqrt{3}/2)ai\) and \(-a/2 - (\sqrt{3}/2)ai\), that are also cube roots of \(n\).

Step 3: Represent the Roots Graphically

Plot these three roots... - Plot the first root (10, 0) as a dot on the positive x-axis. - For the second root, given in form \(a+bi\), \(a\) corresponds to the x-coordinate and \(bi\) to the y-coordinate. Thus, plot the second root (-5,5\(\sqrt{3}\)) somewhere in the second quadrant.- Repeat for the third root (-5, -5\(\sqrt{3}\)) in the third quadrant.

Step 4: Write the Roots in Standard Form

The standard form for a complex number is \(a + bi\). Thus, the cube roots of 1000 in standard form are 10 (or 10 + 0i for consistency), -5 + 5\(\sqrt{3}i\) and -5 - 5\(\sqrt{3}i\).

Key Concepts

Cube Roots

Understanding cube roots is essential in the study of complex numbers. A cube root of a number is a value that, when multiplied by itself three times, gives the original number. For instance, the cube root of 1000 is 10 because \[ 10 \times 10 \times 10 = 1000. \]Cube roots are unique because each real number has exactly three cube roots when considering complex numbers. These roots are located evenly spaced around a circle in the complex plane.

• The principal cube root (real root) is usually straightforward to identify. For 1000, the principal root is 10.
• The two complex roots are derived using the property that if \(a\) is a cube root of \(n\), there are two other related roots: \(-a/2 + (\sqrt{3}/2)ai\) and \(-a/2 - (\sqrt{3}/2)ai\).

Understanding these cube roots can help solve various equations and problems in mathematics. Considering the complex plane adds a layer of depth by visualizing these non-real numbers.

Graphical Representation

Graphical representation is a vital tool for visualizing complex roots. Each complex number corresponds to a point in the complex plane, which gives a clear view of its real and imaginary components. In the example of the cube roots of 1000, we have:

• The first root, 10, sits on the positive x-axis at (10, 0).
• The second complex root, \(-5 + 5\sqrt{3}i\), appears in the second quadrant at (-5, \(5\sqrt{3}\)).
• The third complex root, \(-5 - 5\sqrt{3}i\), is located in the third quadrant at (-5, \(-5\sqrt{3}\)).

Plotting these on a complex plane offers a visual representation that can make understanding their symmetries and distributions easier.
This visualization process also provides insights into the geometry of complex numbers, showing how the roots are evenly distributed in a circular layout. The complex plane turns abstract numbers into geometric figures, aiding comprehension and retention of these concepts.

Standard Form of Complex Numbers

The standard form of a complex number is written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. This form is crucial for performing operations and comparisons involving complex numbers.

• In the context of the cube roots of 1000, we express the results in this standard form:
• 10 translates to \(10 + 0i\), though typically we drop the \(0i\).
• The complex roots \(-5 + 5\sqrt{3}i\) and \(-5 - 5\sqrt{3}i\) are already in standard form.

This form allows for easy addition, subtraction, multiplication, and division of complex numbers.
Moreover, expressing numbers in this way facilitates easier transformations, such as converting to polar form when needed, linking algebraic computation with graphical interpretations.