Problem 49 - Intermediate Algebra

Question

Simplify each expression. Assume that all variables are unrestricted and use absolute value symbols when necessary. See Example 2. $$ \sqrt{a^{4}+6 a^{2}+9} $$

Step-by-Step Solution

Answer

The expression simplifies to \(a^2 + 3\) since \(a^2 + 3\) is always positive.

Step 1: Recognize the Expression Structure

Recognize that the expression inside the square root \( a^4 + 6a^2 + 9 \) is a perfect square trinomial.

Step 2: Identify the Perfect Square Trinomial

Rewrite the expression \( a^4 + 6a^2 + 9 \) in the form of \((a^2)^2 + 2(a^2)(3) + 3^2\), which matches the pattern \((x + y)^2\) where \(x = a^2\) and \(y = 3\).

Step 3: Express as a Binomial Square

Rewrite \( a^4 + 6a^2 + 9 \) as \((a^2 + 3)^2\). This is because \( (a^2 + 3)(a^2 + 3) = a^4 + 6a^2 + 9\).

Step 4: Apply the Square Root

Take the square root of the squared binomial \((a^2 + 3)^2\), which equals \(|a^2 + 3|\), because the square root and square are inverse operations.

Step 5: Simplify Using Absolute Value

Since \(a^2\) is always non-negative, the expression \(a^2 + 3\) is always positive, so \(|a^2 + 3| = a^2 + 3\).

Key Concepts

Understanding Perfect Square Trinomials

A perfect square trinomial is an expression that can be written as the square of a binomial.
This means it takes the form \((x + y)^2\) or \((x - y)^2\).
When expanded, these forms provide us with expressions like \(x^2 + 2xy + y^2\) or \(x^2 - 2xy + y^2\).
In our exercise, we recognized \(a^4 + 6a^2 + 9\) as a perfect square trinomial.
To see this, we can rewrite it to match the pattern \((x + y)^2\).

• First, identify \(x\) as \(a^2\) and \(y\) as 3.
• Next, ensure that it fits in the pattern \(x^2 + 2xy + y^2\).
  For \(a^4 + 6a^2 < + 9\), we get: \( (a^2)^2 + 2(a^2 \cdot 3) + 3^2\).

Once confirmed, we rewrite it as \((a^2 + 3)^2\). This process helps to simplify the original expression later.

Understanding and Forming a Binomial Square

A binomial is a mathematical expression with two terms.
When we square a binomial, meaning multiply it by itself, we get a special form.
For instance, squaring the binomial \((x + y)\) results in the form \(x^2 + 2xy + y^2\).
In the given problem, after identifying a perfect square trinomial, we formed a binomial squared:
\((a^2 + 3)^2\). This matches with our identified pattern where:

• \(x = a^2\)
• \(y = 3\)

This transformation is helpful because when you take a square root of a binomial square, it simplifies
directly to an absolute value form. Thus, this step is crucial in the simplification process.

The Role of Absolute Value in Simplification

Absolute value indicates the distance of a number from zero on the number line, and is always non-negative.
In algebra, it ensures the expression remains positive, even after simplification.
After obtaining \((a^2 + 3)^2\) in our problem, extracting the square root leads to \(|a^2 + 3|\).
The absolute value is crucial because it guarantees the result evaluates a non-negative quantity.
However, since \(a^2\) is always non-negative (as any square is), \(a^2 + 3\) is positive.
Hence, it affirms \(|a^2 + 3| = a^2 + 3\).

• Absolute value is essential when squaring, as squares make quantities positive anyway.
• This ensures even variables constrained under certain conditions maintain appropriate values.

Square Root and Its Simplifying Power

A square root operation "undoes" squaring, helping to simplify expressions.
When you square a number and then immediately apply a square root, you're left with the absolute value of the original number:
This is because only non-negative results occur from taking a square root.
For our calculation, we applied the square root to \((a^2 + 3)^2\):
This brought the expression to \(|a^2 + 3|\).
Here, the square root assists in simplifying the expression:

• It reduces \((a^2 + 3)^2\) to its basic form, ensuring no more unnecessary complexity.
• The inverse relationship between square and square root inherently reduces squared terms.

This emphasizes why understanding the function of the square root in algebra helps in simplifying complex expressions efficiently.