Problem 41 - Elementary and Intermediate Algebra

Question

Simplify each expression. Assume that all variables are unrestricted and use absolute value symbols when necessary. See Example 2. $$ \sqrt{81 h^{4}} $$

Step-by-Step Solution

Answer

The simplified expression is \(9h^2\).

Step 1: Factor Inside the Square Root

First, we need to recognize that the term inside the square root is a product. The expression \(81h^4\) can be expressed as \( (9h^2)^2 \) because \( 9^2 = 81 \) and \( (h^2)^2 = h^4 \).

Step 2: Apply the Square Root Property

Next, apply the square root to the factored expression. According to the square root property \( \sqrt{a^2} = |a| \), we have:\[ \sqrt{(9h^2)^2} = |9h^2| \]

Step 3: Simplify the Absolute Value Expression

We further simplify by considering the absolute value: \[ |9h^2| = 9|h^2| \] Since \(h\) is unrestricted, the absolute value is necessary to ensure the non-negativity in the output, ensuring:\[ 9 |h^2| = 9h^2 \] This step assumes \( h^2 \) is always non-negative, as squares cannot be negative.

Key Concepts

Square Root Property

The square root property is a fundamental concept in algebra. It helps us to simplify equations and expressions involving square roots. When dealing with square roots, one important principle to remember is that for any real number squared, the square root will give us the absolute value of that number. For example, if we have \(a^2\), the square root is written as \(|a|\). This guarantees the output is non-negative.

How do we apply this in practice? Let's say we have \((9h^2)^2\), which is a perfect square inside the square root. When we express it with the square root property, it becomes \(|9h^2|\). This simple transformation follows the rule \(\sqrt{a^2} = |a|\).

The square root property is essential because it reminds us to consider the entire expression's value. This ensures that any operation staying under the square root remains mathematically accurate and all results are represented as non-negative numbers, crucial when working with real numbers.

Absolute Value

Absolute value is a measure of a number's distance from zero on a number line. It has no regard for direction, meaning it always results in a non-negative value. The notation for absolute value is vertical bars around the expression, such as \(|x|\).

When simplifying expressions, incorporating absolute value is key, especially in those with unrestricted variables. With variables like \(h\) in the expression \(9h^2\), understanding the absolute value helps us express \(|9h^2|\) correctly. Since \(h^2\) results in a non-negative number, it naturally becomes part of an equation's output, keeping values in check.

• Ensures every value remains non-negative.
• Prevents negative results from square roots.
• Represents both positive and negative solutions without altering numeric space in the context.

Absolute values, particularly inside equations, are indispensable as they help transform potential misconceptions involving negative results and maintain mathematical consistency.

Factoring

Factoring is a key process in mathematics used to simplify expressions and solve equations. The process involves breaking down complex expressions into simpler components or factors.

In the example \(\sqrt{81 h^4}\), we start by factoring the expression inside the square root. Here, \(81h^4\) is expressed as \((9h^2)^2\). The breakdown occurs as follows:

• 81 can be rewritten as \(9^2\).
• The term \(h^4\) transforms into \((h^2)^2\).

Through factoring, we're restructuring the equation to reveal underlying patterns that can be leveraged in further calculations.

Factoring is not just about breaking down terms; it’s about simplifying as well and enabling the application of other mathematical properties. It acts as a foundation that supports more advanced methods such as finding square roots and applying other algebraic techniques. Understanding the role of factors is critical in simplifying square root expressions and similar equations.