Problem 119 - College Algebra

Question

Which one of the following is true? a. Neither \((-8)^{1 / 2}\) nor \((-8)^{1 / 3}\) represent real numbers. b. \(\sqrt{x^{2}+y^{2}}=x+y\) c. \(8^{-1 / 3}=-2\) d. \(2^{1 / 2} \cdot 2^{1 / 2}=2\)

Step-by-Step Solution

Answer

Option d is the correct choice.

Step 1: Evaluate Option a

Firstly, let's evaluate \((-8)^{1 / 2}\) and \((-8)^{1 / 3}\). Notice that the root of a negative number is an imaginary number when the root is an even number, but a real number when the root is an odd number. Therefore, \((-8)^{1 / 2}\) is an imaginary number, but \((-8)^{1 / 3}\) is -2, which is real. So option a is incorrect because it states neither of them are real.

Step 2: Evaluate Option b

\(\sqrt{x^{2}+y^{2}}=x+y\) is false in general. It would only be true if \(x^{2}+y^{2}=(x+y)^{2}\) which simplifies to \(x^{2}+2xy+y^{2}=x^{2}+y^{2}\). This is a contradiction unless \(xy = 0\), meaning at least one of \(x\) and \(y\) must be zero. Therefore, option b is incorrect.

Step 3: Evaluate Option c

For \(8^{-1 / 3}\), using the property of exponents, it could be changed to \((8^{1 / 3})^{-1}\). The cube root of 8 is 2, therefore it reduces to \(2^{-1}\) which is 0.5. So option c is incorrect.

Step 4: Evaluate Option d

The mathematical operation in option d is \(2^{1/2} \cdot 2^{1 / 2}\). Since \(2^{1/2}\) is the square root of 2, multiplying it by itself is equal to 2. Hence, option d is correct.

Key Concepts

Imaginary Numbers

An imaginary number is a fascinating concept that helps us expand the number system. It is defined as the square root of a negative number. Normally, in real numbers, the square root of a negative number does not exist because there is no real number which, when squared, will give a negative result. This is where imaginary numbers step in.

Imaginary numbers use the unit 'i', which stands for the square root of \(-1 \), or \( i = \sqrt{-1} \). Here are some key points to remember about imaginary numbers:

• They emerge when we try to find the square roots of negative numbers.
• They can be useful for solving equations that have no real solutions.
• In engineering and physics, they are crucial for dealing with wave behavior and electrical circuits.

In our exercise, \((-8)^{1/2}\) is an example of a problem that cannot be solved with real numbers alone. Its solution involves imaginary numbers, as mentioned because it's essentially asking for the square root of a negative number.

Cube Roots

In mathematics, a cube root of a number refers to a value that, when multiplied by itself three times, gives the original number. It is often represented as \( n^{1/3} \).

When it comes to negative numbers and cube roots, there's something interesting to note. The cube root of a negative number is indeed a negative real number.

• This means cube roots allow negative numbers to exist within the real number system, unlike square roots.
• For example, \( (-8)^{1/3} \) gives -2, because \( (-2) imes (-2) imes (-2) = -8 \).
• Cube roots are crucial in many fields, such as solving polynomial equations and analyzing functions in physics.

This concept is important to understanding why option a in our exercise is incorrect, as the cube root of -8 produces a real number.

Real Numbers

Real numbers are all numbers that can be found on the number line. This category includes both rational and irrational numbers. Some of the key points about real numbers are:

• They include numbers like integers, fractions, and decimals.
• Square roots of non-negative numbers are real; however, if it's a negative, we enter into the domain of imaginary numbers.
• Real numbers are used every day in counting, measuring, and quantifying data.

In our exercise, whenever the mathematical operation involves square roots of negative numbers, like \((-8)^{1/2}\), it results in an imaginary number. But \((-8)^{1/3}\) results in a real number. This difference highlights the versatility and completeness of the real number system in combination with the imaginary numbers to provide solutions to a wider range of problems.