Problem 34 - Intermediate Algebra

Question

Simplify each expression. Assume that the variables can be any real number, and use absolute value symbols See Example 2. $$ \left(-16 q^{4}\right)^{1 / 2} $$

Step-by-Step Solution

Answer

The simplified expression is \(4|q^{2}|\).

Step 1: Apply the Square Root to the Coefficient

The problem is \( \left(-16 q^{4}\right)^{1/2} \). To start simplifying, we take the square root of the coefficient \(-16\). Since we are dealing with real numbers, and because the square root of a negative number involves complex numbers, we need to consider \(-16\) as \((-1) \times 16\). Therefore, the square root of \(-16\) is \( \sqrt{-1} \times \sqrt{16} \). However, we will only address the real number part here, that is: \( \sqrt{16} = 4 \).

Step 2: Use Absolute Values for the Variable

Next, we focus on the variable part \(q^{4}\). The square root of \(q^{4}\) is \(q^{2}\). When taking the square root of an even-powered variable term, we use absolute values to ensure we account for both positive and negative roots: \( \sqrt{q^{4}} = |q^{2}| \).

Step 3: Combine Results

Combine the results from previous steps. The absolute value result for \(q^{2}\) and the simplified square root coefficient give us the final expression: \( 4|q^{2}| \). This ensures the expression remains real and correct for all real numbers \(q\).

Key Concepts

Absolute Value

When we simplify expressions involving variables, it is important to handle such that the expression remains valid for all possible values. The absolute value, denoted by vertical bars, helps us achieve this. It transforms both negative and positive numbers into their non-negative counterparts. For example, the absolute value of -5 and 5 is 5.

• Absolute value ensures that the answer is always non-negative.
• It is especially useful when dealing with even powers of variables like in our exercise.
• In the given problem, the expression \(q^4\) becomes \(|q^2|\) when taking its square root, because \(q^2\) could be negative, and using absolute value ensures correctness.
• This is particularly crucial when expressions are under a square root, since square roots of negative numbers are not real.

Understanding when and why to use absolute value symbols helps maintain the integrity of mathematical expressions, keeping them valid under real number assumptions.

Square Root

The square root, symbolized by \(\sqrt{}\), is a fundamental mathematical operation. It answers the question, "What number, when multiplied by itself, equals the given number?" If you consider \(\sqrt{16}\), the answer is 4 because 4 times 4 equals 16.

• Square roots only produce non-negative outputs.
• We often deal with the principal square root, which is the non-negative root.
• In the exercise, the term \(\sqrt{-16}\) was simplified by splitting into \(\sqrt{-1}\) and \(\sqrt{16}\).
• As \(\sqrt{-1}\) involves imaginary numbers, we focus on \(\sqrt{16} = 4\) to stay within real numbers.

This approach ensures that our expression stays in the realm of real numbers while simplifying.

Real Numbers

Real numbers encompass all numbers on the number line, including positive, negative, integers, fractions, and irrational numbers such as \(\pi\) and \(\sqrt{2}\). They are contrasted with complex numbers which include imaginary units, such as the square root of negative one.

• All algebraic operations in this context aim to keep expressions as real numbers.
• In our exercise, we only considered the real portion of \(-16\) by taking \(\sqrt{16}\) while ignoring \(\sqrt{-1}\) because the latter is not a real number.
• When simplifying expressions, always consider if the operation keeps the outcome within the real number system.
• Understanding the limitation of expressions to real numbers is important to prevent unrealistic or erroneous results.

By sticking to operations that yield real numbers, we maintain real-world applicability and relevance.