Problem 120 - Elementary and Intermediate Algebra

Question

$$ \text { a. } \sqrt{-64} \text { b. } \sqrt[3]{-64} $$

Step-by-Step Solution

Answer

a. \( 8i \); b. \( -4 \).

Step 1: Understanding Square Roots of Negative Numbers

When dealing with the square root of a negative number, recognize that the square root of any negative number requires the use of imaginary numbers. The imaginary unit is represented as \( i \), where \( i = \sqrt{-1} \). This allows us to rewrite \( \sqrt{-64} \) as \( \sqrt{64} \times \sqrt{-1} \).

Step 2: Simplifying \( \sqrt{-64} \) Using Imaginary Numbers

Calculate \( \sqrt{64} \), which is 8. Since \( \sqrt{-1} = i \), we can express \( \sqrt{-64} \) as \( 8i \).

Step 3: Understanding Cube Roots of Negative Numbers

Cube roots of negative numbers are real numbers. This is because multiplying a negative number by itself twice results in a positive number, and multiplying again by the same negative number results in a negative number. For instance, calculate \( (-a)^3 = -a^3 \).

Step 4: Calculating \( \sqrt[3]{-64} \)

We need to find a number \( x \) such that \( x^3 = -64 \). Since \( (-4)^3 = -64 \), it follows that \( \sqrt[3]{-64} = -4 \).

Key Concepts

Square Roots of Negative Numbers

When faced with the square root of a negative number, it's important to remember that standard real numbers don't apply. In the field of mathematics, the square root of a negative number would not exist without the concept of imaginary numbers. For example, trying to directly solve \( \sqrt{-64} \) using only positive or negative real numbers will lead to confusion.
Imaginary numbers offer a solution by introducing a new component denoted by the imaginary unit \( i \), where \( i = \sqrt{-1} \). So, when tackling \( \sqrt{-64} \), you first break it down into \( \sqrt{64} \) and \( \sqrt{-1} \).
Here's how it works:

• Find \( \sqrt{64} \), which is a straightforward 8. This part remains a real number.
• Recognize \( \sqrt{-1} \) as the imaginary unit \( i \).
• Combine them to express the square root of the negative number: \( \sqrt{-64} = 8i \).

When you express the square root of a negative number using \( i \), you're effectively transitioning into the realm of complex numbers, where solutions become possible.

Cube Roots of Negative Numbers

Unlike the square roots, cube roots of negative numbers remain within the real number system. This is because a negative number, when cubed, cycles through positive and negative intermediate results, ultimately returning to a negative result.
To solve \( \sqrt[3]{-64} \), we need to find a real number that, when cubed, equals the original negative value. Interestingly, multiplying three wholes of a negative number yields a negative outcome:

• Calculate the cube root of \(-64\) by determining which number cubed will equal \(-64\).
• Notice that \((-4)^3 = -64\), through the operation: \((-4) \times (-4) \times (-4)\) results in \(-64\).

This process shows that cube roots of negative numbers don't need imaginary numbers like square roots do, because the real number domain accommodates them naturally.

Imaginary Unit i

The imaginary unit, denoted as \( i \), is a mathematical construct that allows us to extend our number system beyond the real numbers and tackle equations that otherwise would have no real solution.
By definition, \( i \) is equal to the square root of \(-1\) or \( i = \sqrt{-1} \). This might seem abstract at first, but it holds a specific purpose: to simplify the square roots of negative numbers where typical operations break down.

• Using \( i \), complex numbers are formed as a combination of real numbers and imaginary numbers, e.g., \( a + bi \).
• This can be quite useful in engineering, physics, and other disciplines where variables may oscillate between states.

Imaginary numbers, through \( i \), open the door to an entire complex plane where every number is expressed as a point with a real and an imaginary part. Consequently, \( i \) is crucial for expanding mathematics into new dimensions.